THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES JUE on c below MiG SOUTHERN BRANCH DNIVPR-ITY OF CALIFGRNI^ I IBRARY, \ u.(y'>/2c^...^^6^ DIFFERENTIAL EQUATIONS. PRINTED BY C. J. CLAY, M.A. AT THE UNU'EESITY PRESS. A TREATISE ON DIFFERENTIAL EQUATIONS. BY GEORGE BOOLE, F.RS. PROFESSOR OP 3IATHEJIATICS IN THE QUEEX'S UXm:RSITY, IREL.OCD, HONORAEY MEMBER OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. SECOND EDITION, REVISED. CTambri&ge antJ Hontion : MACMILLAN AND CO. 1865. [The right of Tratislaiion is reserved] r63 C ,C, tC £ En-g^neer'ms 9k MathcmjtICil ^l PKEPACE. I HAVE endeavoured, in tlie following treatise, to convey vy as complete an account of the present state of knowledge on the subject of Differential Equations, as was consistent witli ^ the idea of a work intended, primarily, for elementary instruc- tion. It was my object, first of all, to meet the wants of those who had no previous acquaintance with the subject, but I also desired not quite to disappoint others who might seek for more Kadvanced information. These distinct, but not inconsistent I aims determined the plan of composition. The earlier sections V of each chapter contain that kind of matter which has usually ^ been thought suitable for the begnmer, while the latter ones are devoted either to an account of recent discovery, or to the discussion of such deeper questions of principle as are likely to present themselves to the reflective student in connexion with the methods and processes of his previous course. An appen- dix to the table of contents will shew what portions of the work are regarded as sufficient for the less complete, but still not unconnected study of the subject. The principles which I have kept in view in carrying out the above design, are the following : 1st, In the exposition of methods I have adhered as closely as possible to the historical order of their development. I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary — being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had VI PllEFACE. arrived. And tliese are causes wliicli operate in perfect har- mony. Each new scientific conception gives occasion to new applications of deductive reasoning; but those applications may be only possible through the methods and the processes which belong to an earlier stage. Thus, to take an illustration from the subject of the follow- ing work, — the solution of ordinary simultaneous differential equations properly precedes that of linear partial difierential equations of the first order; and this, again, properly precedes that of partial differential equations of the first order which are not linear. And in this natural order were the theories of these subjects developed. Again, there exist large and A^ery important classes of differential equations the solution of which depends on some process of successive reduction. Now such reduction seems to have been effected at first by a repeated change of variables ; afterwards, and wdth greater generality, by a combination of such transformations with others involv- ing differentiation ; last of all, and with greatest generality, by symbolical methods. I think it necessary to direct attention to instances like these, because the indications which they afibrd appear to me to have been, in some works of great ability, overlooked, and because I wish to explain my motives for departing from the precedent thus set. Now there is this reason for grounding the order of exposi- tion upon the historical sequence of discovery, that by so doing Vv^e are most likely to present each new form of truth to the mind, precisely at that stage at which the mind is most fitted to receive it, or even, like that of the discoverer, to go forth to meet it. Of the many forms of false culture, a premature converse with abstractions is perhaps the most likely to prove fatal to the growth of a masculine vigour of intellect. In accordance with the above principles I have reserved the exposition, and, with one unimportant exception, the ap- plication of symbolical methods to the end of the work. The PEEFACE. VU propriety of tills course appears to mc to be confirmed by an examination of the actual processes to which symbolical methods, as applied to differential equations, lead. Generally speaking, these methods present the solution of the proposed equation as dependent upon the performance of certain inverse operations. I have endeavoured to shew in Chap, xvi., that the expressions by which these inverse operations are symbol- ized are in reality a species of interrogations, admitting of answers, legitimate, but differing in species and character ac- cording to the nature of the transformations to which the expressions from which they are derived have been subjected. The solutions thus obtained may be particular or general, — they may be defective, wholly or partially, or complete or redundant, in those elements of a solution which are termed arbitrary. If defective, the question arises how the defect is to be supplied ; if redundant, the more difficult question whether the redundancy is real or apparent, and in either case how it is to be dealt with, must be considered. And here the necessity of some prior acquaintance with the things themselves, rather than with the symbolic forms of their ex- pression, must become apparent. The most accomplished in the use of symbols must sometimes throw aside his abstrac- tions and resort to homelier methods for trial and verification — not doubting, in so doing, the truth which lies at the bottom of his symbolism, but distrusting his own powers. The question of the true value and proper place of symboli- cal methods is undoubtedly of great importance. Their con- venient simplicity — their condensed power — must ever consti- tute their first claim upon attention. I believe however that, in order to form a just estimate, we must consider them in another aspect, viz. as in some sort the visible manifestation of truths relating to the intimate and vital connexion of language with thought — truths of which it may be presumed that we do not yet see the entire scheme and connexion. But, VI 11 PREFACE. wliile tliis consideration vindicates to them a high position, it seems to me clearly to define that position. As discussions about words can never remove the difficulties that exist in things, so no skill in the use of those aids to thought w^hich language furnishes can relieve us from the necessity of a prior and more direct study of the things which are the subjects of our reasonings. And the more exact, and the more com- plete, that study of things has been, the more likely shall we be to employ with advantage all instrumental aids and appliances. But although I have, for the reasons above mentioned, treated of symbolical methods only in the latter chapters of the work, I trust that the exposition of them which is there given will repay the attention of the student. I have endea- voured to supply what appeared to me to be serious defects in their logic, and I have collected under them a large number of equations, nearly all of which are important, — from their con- nexion with physical science or for other reasons. 2ndly, I have endeavoured, more perhaps than it lias been usual to do, to found the methods of solution of differential equations upon the study of the modes of their formation. In principle, this course is justified by a consideration of the real nature of inverse processes, the laws of which must be ulti- mately derived from those of the direct processes to which they stand related ; in point of expediency it is recommended by the greater simplicity, and even in some instances by the greater generality, of the demonstrations to which it leads. I would refer particularly to the demonstration of Monge's method for the solution of partial differential equations of the second order given in Chap. xv. With respect to the sources from which information has been drawn, it is proper to mention that, on questions re- lating to the theory of differential equations, my obligations are greatest to Lagrange, Jacobi, Cauchy, and, of living PEEFACE TO THE SECOND EDITION. In composing his Treatise on Differential Equations Pro- fessor Boole found himself deeply interested in the subject to which his first labours as an original investigator had been devoted. In consequence he determined soon after the publication of the volume to continue his studies and re- searches with the design of ultimately reconstructing the Treatise on a more extensive scale. During the last six years of his life he worked steadily at this object; and he was about to send the first sheets of the new edition to the press when he was attacked by the illness which terminated in his sudden and lamented death. His manuscripts were entrusted to me early in the present year. After careful consideration it seemed to me that the best plan to pursue was to reprint the original volume, and to collect into a supplementary volume the additional matter which had been prepared for enlarging the work. The pro- priety, I might almost say the necessity, of this course will be shewn more conveniently in the preface to the supple- mentary volume, which will soon be published. The present volume then is a reprint of the original Treatise with changes and corrections, some of which were indicated in Professor Boole's interleaved copy, and some of which have been made on my own authority. The sheets have been carefully read by the Eev. J. Sephton, Fellow of St John's College, as well as by myself; and I trust that few misprints or errors will now be found in the volume. I. TODHUXTER. St John's College, Casibkidge, October, 1 86?. PREFACE. IX writers, to Professor De Morgan. For metliods and exam- ples, a very large number of memoirs English and foreign have been consulted : these are, for the most part, acknow- ledged. At the same time it is right to add that, in almost every part of the work, I found it necessary to engage more or less in -original investigation, and especially in those parts which relate to Eiccati's equation, to integrating factors, to singular solutions, to the inverse problems of Geometry and Optics, to partial differential equations both of the first and second order, and, as has already been intimated, to symboli- cal methods. The demonstrations scattered through the work are also many of them new, at least in form. In recent years much light has been thrown on certain classes of differential equations by the researches of Jacobi on the Calculus of Variations, and of the same great analyst, with Sir W. E. Hamilton and others, on Theoretical Dy- namics. I have thought it more accordant with the design of an elementary treatise to endeavour to prepare the way for this order of inquiries than to enter systematically upon them. This object has been kept in view in the writing of various portions of the following work, and more particularly of that which relates to partial differential equations of the first order. GEOEGE BOOLE. Queen's College, Cork, February, 1859. CONTENTS. CHAPTER I. PAGE OF THE NATURE AND ORIGIN OF DIFFERENTIAL EQUATIONS 1 Definition, 2. Species, Order and Degree, 3. General Solution, Com- plete piimitive, 6. Genesis of Differential equations, 8. How- many of each order posbible, — how many independent, 15. Cri- terion of derivation from a common primitive, 16. General form of equations thus related, 1 7. Geometrical illustrations, 1 8. Ex- ercises, 20. CHAPTER II. ON DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND DEGREE BETWEEN TWO VARIABLES . General equation Mdx + Xchj = 0, 23. Complete pi-imitive / {x, y) = c, 26. Geometrical illustration, 28. Various integrable cases, 29. Homo- geneous equations, 34. Linear equations of first order, 38. Other forms, 40. Solution by development, 41. Exercises, 44. CHAPTER III. EXACT DIFFERENTIAL EQUATIONS OF THE FIRST DEGREE 47 General criterion, 47. INIode of solution, 50. Practical simplifications, 52. Exercises 53. XU CONTENTS. CHAPTER lY. PAGE ON THE IXTEGRATIXG FACTORS OF THE DIFFEllEXTIAL EQUATION 2Idx + Ndy = Q . . . 55 Integrating factors always exist, 56. General form of integrating factors, 57. Special detei-minations for homogeneous equations and for equations of the iorm. Fi {xy)7/clx + F2{xy) xdy = o, 59 — 62. Exercises, 67. CHAPTER V. ON THE GENERAL DETER:yiINATION OF THE INTEGRATING FACTORS OF THE EQUATION Mdx + Ndy = 0. . 69 Partial differential equation for integrating factors, 69. Solution when the integrating factor is a function of x, 70 — of?/, 71 — oixy, 72. When homogeneous, 74 — 76. More general application, 82. So- lution of P-^dx + PcAy + Q {xdy - ydx) = 0, 84 . Jacobi's equation, 85 . Euler's method, 86. Exercises, 88. CHAPTER YI. ON SOJIE REMARKABLE EQUATIONS OF THE FIRST ORDER AND DEGREE 91 Riccati's equation ^ + Jm^ _ (,y^m.^ ^ j _ x -- — ay + hy" = cx^, 92 . Solution by continued fractions, 96, 97. Kiccati's equation made linear, 103. Euler's equation, 104. Theoremof development, 107. Exercises, no. CHAPTER YII. ON DIFFERENTIAL EQUATIONS OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE . . .113 Typical form, 113. Theory of its solution, 115. Relation of complete primitive to particular integrals, &c., 117. Special methods, 121. One variable only involved, 122. X(p (2^) + y\p {p) = x{p), i'24. Clairaut's equation, ib. Singular solutions, 125. Homogeneous equations, 128. Equations solvable by differentiation, 131. Trans- formations, 134. Exercises, 136. CONTENTS. XI 11 CHAPTER YIII. PAGE OX THE SINGULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. . . .130 Primary definition — positive and negative marks, 139, 140. Derivation of the singular solution from the complete primitive, 141, General theorem, 148. Geometrical interpretation, 152. Derivation of the singular solution from the differential equation, 153. More o-eneral definition of singular solution, 163. Distinction of species, ib. General theorem, 164. Eeview of prior methods, 174. Properties of singular solutions, 177. Exercises, 182. CHAPTER IX. ON DIFFERENTIAL EQUATIONS OF AN ORDER HIGHER THAN THE FIRST . . . .187 Relation to complete primitive, 187. Solution by development, 189. Linear equations, 192 — 199. General rule when the second member is o, 200. Second member a function of x, — Variation of para- meters, ib. Dependent class, 203. Properties of linear equa- tions, 204. Analogy with Algebraic equations, 206. Exercises, 207. CHAPTER X. EQUATIONS OF AN ORDER HIGHER THAN THE FIRST, CONTINUED 209 One variable wanting, 209. One differential coefficient present, 2ir. dx^%lx--0'"'"' d^^=Ad^')'"''' ^'"^'Seneousequ^. tions, 215. Exact equations, 222. Miscellaneous methods and ex- amples, 226. Singular integrals, 229. Exercises, 234. CHAPTER XL GEOMETRICAL APPLICATIONS. . . .238 Different problems, 239 — 244. Trajectories, 245. Curves of Pursuit, 25 1. Solution of a differential equation, 254. Involutes, 256. Inverse problem of caustics, 258 — 263. Direct problem, 259. Intrinsic equation of a curve, 263. Exercises, 269. XIV CONTENTS. CHAPTER XII. 1^ PAGE ORDINARY DIFFERENTIAL EQUATIONS WITH MORE THAN TWO VARIABLES . . . .272 Meaning oi Pdx+Qdi/ + Rdz = o, 2 72. Condition of derivation from a single primitive, 275. Solution, 276. General rule and ex- amples, 279. Homogeneous equations, 281. Integrating factors, 282. Equations not derivable from a single primitive, 283. More than three variables, 286. Equations of an order higher than the first, 289. Exercises, 291. CHAPTER XIII. SIMULTANEOUS DIFFERENTIAL EQUATIONS. . 292 Meaning of a determinate system, 292. General theory of simultaneous equations of the first order and degree, 293 — 307. Systems of two equations, 294. Of more than two, 298. Linear equations with constant coefficients, 300. Equations of an order higher than the first, 307. Exercises, 316. CHAPTER XIY. OF PARTIAL DIFFERENTIAL EQUATIONS . .319 Nature, 319. Primary modes of genesis, 321. Solution when all the differential coefficients have reference to only one of the independent variables, 322. Linear equations of first order, 324. Their genesis, 325. Their solution, 329. Non-linear equations of the first order, 335. Complete primitive, general primitive, and singular solution, 339. Sufficiency of a single complete primitive, 345. Singular solutions, 346. Geometrical applications, 347. Symmetrical and more general solution of equations of the first order, 350. Exercises, 358. CHAPTER XY. PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 361 The eqn&iion lir+Ss + Tt—V, 361. Condition of its admitting a first integral of the form u—f{v), 362. Deduction of such integral when possible, 364. Relations of first integrals, 366. General rule, 369. jVIiscellaneous theorems. Poisson's method, 375. Duality, 376. Legendre's Transformation, 379. Exercises, 380. CONTENTS. XV CHAPTER XYT. PAGK SYMBOLICAL METHODS . . .381 Laws of direct expressions, 381 — 384. Inverse forms, 385. Linear equations with constant coefficients, 388. Forms purely symbolical, 398. Equations solvable by means of the properties of homogeneous functions, 403. The method generalized, 406. Exercises, 4 10. CHAPTER XYII. SYMBOLICAL METHODS, CONTINUED . .412 Symbolical form of differential equations with variable coefficients, 412. Finite solution, 415 — 436. Reduction of binomial equations, 418. Pfaff's equation, 430. Equations not binomial, 432. Solution by series, 437. Evaluation of series, 44 1 . Generalization, 446. Theo- rem of development, 447. Laplace's reduction of partial differential equations, 450. Miscellaneous notices, 454. Exercises, 457. CHAPTER XVIII. SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS BY DEFINITE INTEGRALS . , . 4G1 Laplace's method, 461. Partial differential equations, 475. Parseval's theorem, 477. Solution by Fourier's theorem, 478. Miscellaneous exercises, 481. The follomng portions of the icorh are recommended to leglnncrs. Chap. I. Arts, r— 8, 11. Chap. II. Arts. 1— 11. Chap. III. Chap. IV. Arts. 1—9. Chap. V. Arts. 1—4. Chap. VI. Arts, i — 10. Chap. VII. Arts. 1—8. Chap. VIII. Arts. 1—7. Chap. IX. Arts, t, 3—12. Chap. X. Arts, i — 3. Chap. XI. Arts, i — 7. Chap. XII. Arts. 1—8. Chap. XIII. Chap. XIV. Arts. 1—12. Chap. XV. Arts. 1—8. ,Chap. XVI. Arts. 1—8. Chap. XVII. Arts, r — 12. Chap. XVIII. Arts. 1—8. DIRECTION TO BINDER. Plate to face page 271. DIFFERENTIAL EQUATIONS. CHAPTEE I. OF THE NATURE AND ORIGIN OF DIFFERENTIAL EQUATIONS. 1. What is meant bj a differential equation ? To answer tliis question we must revert to tlie fundamental conceptions of the Differential Calculus. The Differential Calculus contemplates quantity as subject to variation ; and variation as capable of being measured. In comparing any two variable quantities x and ?/ connected by a known relation, e.g. the ordinate and abscissa of a given curve, it defines the rate of variation of the one, y, as referred to that of the other, x, by means of the fundamental con- ception of a limit ; it expresses tliat ratio by a differential coefficient ~ ; and of that differential coefficient it shews how dx to determine the varying magnitude or value. Or, again, con- siderins; -f^ as a new variable, it seeks to determine the rate ° ax of its variation as referred to the same fixed standard, the variation of x, by means of a second differential coefficient -T*^ , and so on. But in all its applications, as well as in its theory and its processes, the primitive relation between the variables x and y is supposed to be known. In the Integral Calculus, on the other hand, it is the rela- tion among the primitive variables, x, y, &c. which is sought. In that branch of the Integral Calculus with which the student ^ B. D. E. 1 2 OF THE NATURE AND ORIGIN [CH. I. 13 supposed to be already familiar, tlie differential coefficient — being given in terms of tlie independent variable a?, it is die proposed to determine tlie most general relation between y and X. Expressing the given relation in tlie form f>*(-) «' the relation sought is exhibited in the form i/ = l(f){x) dx + c. In (1) we have a particular example of an equation in the expression of which a differential coefficient is involved. But (221 instead of having as in that example -,- expressed in terms of £c, we might have that differential coefficient expressed in terms of ?/, or in terms of x and y. Or we might have an equation in which differential coefficients of a higher order, -^4 , -ts ? &c., were involved, with or without the primitive variables. All these including (1) are examples of differential equations. The essential character consists in the presence of differential coefficients. The equations d"^!! dif ^ , . are seen to be differential equations, the latter of which con- tains, while the former does not contain, the primitive vari- ables. And thus we are led to the following definition. Def. a differential equation is an expressed relation in- volving differential coefficients, ivith or icitliout tlie primitive cariables from which those differential coefficients are derived. ART. 2.] OF DIFFERENTIAL EQUATIONS. 3 That which gives to the study of differential equations its peculiar value, is the circumstance that many of tlic most im- jjortant conceptions of Geometry and Mechauics can only be realized hi thought by means of the fundamental conception of the limit. When such is the case, the only adequate ex- ])ression of those conceptions in language is through the me- dium of differential coefficients, — the only adequate expression of the truths and relations of which they are the subjects is in the form of differential equations. Species, Order and Degree. 2. The species of differential equations are determined either by the mode in which differential coefficients enter into their composition, or by the nature of the differential coeffi- cients themselves. We may thus distinguish two great pri- mary classes of differential equations, viz. : 1st. Ordinary differential equations, or those in which all the differential coefficients involved have reference to a single independent variable. 2ndly. Partial differential equations, characterized by the presence of 'partial dafferential coefficients, and tlierefore in- dicating the existence of two or more independent variables with respect to which those differential coefficients liave been formed. Thus an equation such as (2) or (3), involving no other differential coefficients than ~j- , -^, &c. is an ordinary dif- aju ajc ferential equation, in which x is the independent, y the de- pendent variable. An equation involving -^ and -y^ would, ax a u on the contrary, be a partial differential equation, having ;:; for its dependent, x and y for its independent variables. The dz dz . . 1 T^ equation a; j +y . = ;3; is a partial differential equation. dx "^ dy ^ ^ ^ The^ present chapter will be chiefly devoted to the con- sideration of that class of ordinary differential equations in 1—2 4 SPECIES, ORDER AND DEGREE. [CII. I. whicli there exists a single independent variable x, a single dependent variable ?/, and one or more of the differential coefficients of y taken with respect to x ; the presence of the last element only, viz. the differential coefficient, being essen- tial (Art. 1). The two following equations, in addition to those already given, will exemplify some of the chief varieties of the species under consideration : dv A . 1 + f^T^ (i). {'-(DT d^ = mx (5). In (4) the independent variable x, the dependent variable y, and the differential coefficient -~- are all involved; but, while in the previous examples ~ appears only in the first degree, in the present one it appears in the second degree and under a radical sign. In (5) we meet with the second dh/ differential coefficient -y'^o in addition to the first differential dx' coefficient -j- and the independent variable x. The typical or general form of a differential equation of the species just described is 4'^'2'g.-S)='> («)' with the condition, already referred to, that one at least of tlie differential coefficients must explicitly present itself. All the above equations may at once be referred to the typical form by transposition of their second member. ART. 3.] SPECIES, ORDER AND DEGREE. 5 3. Differential Equations are ranked in order and degree according to the following principles. 1st. The order of a differential equation is the same as the order of the highest differential coefficient which it con- tains. 2ndly. The degree of a differential equation is the same as the degree to which the differential coefficient which marks its order is raised, that coefficient being supposed to enter into i^iiQ equation in a rational form. Thus the equation \dx/ dx ' is of the first order and of the second degree. The equation d'']/ dif ,2 is of the second order and of the first degree. The equation J = V^(y-^3 (')' reduced to the rational form ;2y-i-^- ••(«)' is seen to be of the first order and second degree. The ground of the preference wliich is to be given to rational forms in the expression and in the classitication of differential equations is, that a rational form is at the same time the most general form of which an equation is sus- ceptible. Thus (8) includes both the equations whicli would be formed by giving different signs to the radical in (7). The typical form of an ordinary differential equation of the first order is evidently ■ /(-'^'S=^ (^)- 6 SPECIES, ORDER AND DEGREE. [CH. I. 4. When a differential equation is capable of being ex- pressed in the form in which the coefficients X^, X^,..,X^ and the second member X are either constant quantities or functions of the indepen- dent variable x only, the equation is said to be linear. Equa- tions (1), (2) and (3) are thus seen to be linear, but (4) and (5) are not linear. If we refer (3), after dividing both members by x^, to the general form (10), we have 71 = 2, A>^, A> ^;,, AVhen the coefhcients X^, X^, &c. in the first member of a linear differential equation referred to the above general type are constant quantities, the equation is defined as a linear differential equation with constant coefficients. When those coefficients are not all constant it is deiined as a linear dif- ferential equation with variable coefficients. The distinction is illustrated in the following examples: ov d II ch] X ) ~~ — x-:--\- 4?/ = cos X, dx dx ^ the former of which is a linear differential equation with constant coefficients, while the latter would be described as a linear differential equation with variable coefficients. Meaning of the terms ^ general solution,'' '' comjjlete 'primitive^ 0. In all differential equations there is, as has been seen, an implied reference to some relation among variable quantities dependent and independent; such reference being established through the medium of differential coefficients. Now the chief object of the study of differential equations is to enable us to ART. 5.J GENERAL SOLUTION. 7 determine whenever it is possible, and in tlie most general manner which is possible, such implied relation among the primitive variables. That relation, when discovered, is, by the adoption of a term primarily applicable to the mode or process of its discovery, called the solution of the equation. Thus if the given equation be x-j^ + y=cosx (11), tlie following process of solution may be adopted. Zdultiply- ing by dx, we have xd?/ + ydx = cos xdx., and Integrating, since each member is an exact differential, cc?/ = since + c (12). The result is termed the solution, or, still more definitely, the general solution of the equation. It involves an arbitrary constant, c, by giving particular values to which a series of particular solutions is obtained. The equations xy — sin a?, xy = since + 1, are particular solutions of the given differential equation. The term solution is still employed, even when the inte- gration necessary in order to obtain in a finite and explicit form the relation between the variables cannot be eft'ected. Thus if we had the differential equation, ^%-y-'-''=^ c^)' we should thence derive in succession xdy — ydx _ e^dx ^ = f^^ + c (U), x J x 8 GENESIS OF DIFFERENTIAL EQUATIONS. [CH. I. and the last result is called the solution of the given equation, although it involves an integration which cannot be performed in finite terms. The relation among the variables which constitutes the general solution of a differential equation, as above described, is also termed its complete primitwe. The relation (14) in- volving the arbitrary constant c is virtually the complete primitive of the differential equation (13). It will be observed that the terms ' general solution ' and ' complete primitive,' though applied to a common object, have relation to distinct processes and to a distinct order of thought. In the strict application of the former term we contemplate the differential equation as prior in the order of thought, and the explicit relation among the variables as thence deduced by a process of solution ; while in the strict use of the latter term the order both of thought and of process is reversed. Genesis of Differential Uquatioiis, 6. The theory of the genesis of differential equations from their primitives is to a certain extent explained in treatises on the Differential Calculus, but there are some points of great importance relating to the connexion of differential equations thus derived, not only with their primitive, but with each other, Aviiich need a distinct elucidation. Suppose that the complete primitive expresses a relation between x, y and an arbitrary constant c. Differentiating on the supposition that x is the independent variable, we obtain a new equation which must involve -— , and which may involve any or all of the quantities x, y and c. If it do not involve c, it will constitute the differential equation of the first order corresponding to the given primitive. If it involve c, tlien the elimination of c between it and the primitive will lead to the differential equation in question. Thus if the complete primitive be ^ = ca; (1), ART. 6.] GENESIS OF DIFFERENTIAL EQUATIONS. 9 we have on differentiation, i- c^)' and, eliminating tlie constant c, y-^% (^)' tlie differential equation of the first order of which (1) is the complete primitive. That primitive might have been so prepared as to lead to the same final equation bj mere differentiation. Thus, re- ducing the primitive to the form ■■^ = 0, X we have on differentiating and clearing the result of fractions, which agrees with (3) . And generally, if a primitive involving an arbitrary constant c be reduced to the form <^ (ic, y) = c, the corresponding differential equation will be obtained by mere differentiation and removal of irrelevant factors, i. e. of fiictors which do not contain — , and do not therefore affect the relation in which --,- stands to x and ?/. For it is. in that dx ^ relation, as already intimated. Art. 2, that the essential character of the differential equation consists. It is to be observed that when the differentiation of a primi- tive involving an arbitrary constant c does not of itself cause that constant to disappear, the result to which it leads is still a differential equation, only not that difierential equation of which the equation given constitutes the complete primitive. Thus, while the complete primitive of (3) is (1), that of (2) is y — cx-\-G^ c being now the arbitrary constant, — arbitrary as being independent of anything contained in the difierential 10 SECOND AND HIGHER OEDERS. [CH. T. equation. Indeed when we consider -^- = c as the differential equation, tlie constant c, as entering into its complete primitive, y = cx-\- c', is not arbitrary, the value which it bears in the primitive being determined by that which it bears in the differential equation. As another illustration of the same theory, the equation y — ce"^ as complete primitive gives rise to the differential equation of the first order while the equation immediately derived from it by differ- du entiation, viz. -j- — c«e'''', has for its complete primitive y = ce'''-\- c . To the last mentioned differential equation, y = ce'''' stands in the relation of a particular primitive. Second and Higher Orders, 1. It is shewn in the previous section that from an equa- tion containing x and y with an arbitrary constant c, we can by differentiation, and elimination (if necessary) of that con- stant, obtain the differential equation of the first order, of which the given equation constitutes the complete primitive. In like manner an equation connecting a?, y, and two arbitrary constants being given, if we differentiate twice, and eliminate, should they not have already disappeared, the arbitrary constants, we shall arrive at a differential equation of the second order free from both the constants in question, and of which the given equation constitutes the complete primitive. Thus, if we take as the primitive equation y = ax^ + I)x (4), ART. 7.] SECOND AND HIGHER ORDERS. 11 we find on difierentiation %^^ax^l (5), and, eliminating h between these equations, dif dx (6), a differential equation of the first order free from the constant h. Differentiating this equation we have d^y _ dx and, eliminating a between the last two equations, "" dx' ^"^ dx^ ^y ^ ^^' a differential equation of the second order free from both a and h. In the above example the constant h was eliminated after the first differentiation, and the constant a after the second. But the same final result would have been arrived at if the order of the eliminations had been reversed. Thus, if a be eliminated between (4) and (5), we shall have x^4- + lx-2y=^0, dx •^ a differential equation of the first order, different in foiTn from (6), and involving h instead of a. But on differentiating this equation and eliminating Z>, we shall arrive at the same final equation of the second order (7). And generally the order in icliicli the constants are eliminated does not affect the form of the final differenticd e, must leave a single final relation between the remaining four, x, y, -~ ^ -j\. And this is the differential equation in question. As another example, let us eliminate the arbitrary constants c and c from the equation y=ce^--fcV^ (8). AET. 7.] SECOND AND IIIGIIER OEDERS. 13 Differentiatli'": we have dy dx ace + Z*cV^ (9). To eliminate c subtract from this equation the primitive (8) multiplied by a ; we have '^£-ay = {h-a)di' (10). Again, differentiating dx' dx ^ ^ ' and (to eliminate c) subtracting from this the previous equa- tion multiplied bj h, we have g_(„ + J)| + ,,, = 0 (11), the differential equation of the second order required. If Ave had first eliminated c we should in the place of (10) have obtained the equation ^£-hj=.{a-h)ce" (12). Differentiating this and eliminating c we again obtain the same final result (11). That result is a differential equation of the second order, and (8), involving both the arbitrary constants c and c, is its com- plete primitive. The intermediate equations (10) and (12), each of which contains one of the arbitrary constants, and from each of which, by the elimination of that constant, the final differential equation may be derived, are \X?> first integrals. As the term primitive has reference to the direct processes of differentiation, &c. by which a differential equation is formed, the term integral has reference to the inverse process of integration by which we reascend from a differential equation to its primitive. Considered witli reference to these processes the primitive is sometimes termed i]iQ final integral. 14 SECOND AND HIGHER ORDERS. [CH. T. It has been shewn that the order of suci^ession in which arbitrary constants are eliminated is indifferent. It may be added, and upon the same gromid, that the elimination may be simultaneous. If we write the primitive (8) in the form y-ce^^ -de'^-. = 0, and differentiate it twice, we have dy -ace"'- ■hce'' = ■0, dhf dx' -a'ce''- - h'de'^ = 0, and, from the above system of three equations eliminating the constants c and c by the method of cross-multiplication, we again arrive at the final differential equation of the second order (11). 8. The above examples prepare us for the general state- ment of the theory of the genesis of differential equations. Let F(x, ?/, Cj , Cg , . . . c„) = 0 be a primitive equation between X and ?/ involving 7i arbitrary constants c^, c^,...c„. Differen- tiating with respect to x, and regarding ?/ as a fanction of x, we obtain directly, or by elimination of c^, an equation of the first order of the form *i^^''^'^' ^-' ^3' •••0 = 0. Differentiating this equation with respect to x, and regarding dy y and -j- as functions of that quantity, we obtain directly, or by elimination of Cg, an equation of the second order of the general form . dy d'^y . Continuing the process, we arrive at a final result of the form ^'*r'^' dx' dj"-djf) "• V .liT. 8.] SECOND AND HIGHER ORDERS. 15 Xow this Is the type of an ordinary differential equation of the ^^'^ order, (6), Art. 2. As, in the above process of differentiation and elimination, we might have begun by eliminating any other of the con- stants instead of c^, it follows that to a primitive containing n arbitrary constants there belong n differential equations of the first order, each involving n — 1 arbitrary constants. But as those differential equations are all formed by mere pro- cesses of elimination from two equations, viz. from the primi- tive and its first derived equation, two only of them are independent. Again, as the differential equations of the second order are formed by eliminating two of the constants Cj , c^ , . . . c„ , and as from n constants, n — - — combinations of two constants can be selected, it is seen that there will exist n — - — differential equations of the second order, each A containing n — 2 arbitrary constants. Of these equations three only will however be independent, the whole system being derived actually or virtually from the primitive and its lirst and second derived equations; — actually if we differen- tiate twice before eliminating; virtually if each differentiation is followed by the elimination of a constant. This process of deduction continued leads to the following ' general theorems, viz. : 1st. To a given jjrimitive involving x, y, and n arbitrary , , T. 7z(n- 1) (71-2) ... (w-r +1) ,.^ ., constants belonq — ^ , ^- ^ '- differential eqiiations of the /^ order {r being any whole number less than 11), each involving n — r arbitrary constants, but of those e<2 tions r + 1 only will be independent. ua- 2nd. There ivill exist one differential equation of the ?i*^ order free from arbitrary cotistants. The converse of the latter truth, viz. that a differential equation of the 7i^^ order implies the existence of a complete primitive involving n arbitrary constants, will be established in a future page. 16 CRITERION OF DERIVATION [CH. Criterion of derivation from a common primitive, 9. It is establislied in Art. 7, 1st, that from a primitive equation involving two arbitrary constants arise two differen- tial equations of the first order, each involving one of those constants; 2ndly, that each of these differential equations of the first order gives rise to the same differential equation of the second order, of which the original equation constitutes the complete primitive or final integral. The second of the properties above noted constitutes a criterion by which it may be determined whether two dif- ferential equations of the first order, each involving an arbi- trary constant, originate from the same primitive. We must differentiate each equation, and then eliminate its arbitrary constant. If the two results agree as differential equations of d'^Tj the second order, i. e. if they give the same value of -y^^ as a function of x, ?/, and y- , the differential equations of the first order must have originated in the same primitive. Fur- dii thermore, that primitive will be obtained by eliminating -~ between the two differential equations given. Ex. The differential equations of the first order ^ J-«^ = o w. ^-^yt-' (^)' are both derived from the same primitive. Each of them leads on differentiation and elimination of its arbitrary con- stant to the differential equation of the second order, ^3.2.^(|)-.| = 0 (3). ART. 10.] FROM A COMMON PRIMITIVE. 17 The primitive, found by eliminating — from tlie given equations, is f-a-i^l (4), a and h being arbitrary constants. 10. The differential equations of the first order which constitute the first integrals (Art. 7) of a differential equation of the second order (as, in the above example, (1) and (2) are first integrals of (3)), may by algebraic solution be reduced to the forms ■^("■2/. JS=« ^•5)- ^(-'^'i)=^ •(«)■ Xow a function of the arbitrary constants a and Z>, as ^(a, V)^ is itself an arbitrary constant, and may be represented by c. Hence any equation of the form dxr ^ V '-^^ dx would, equally with (o) and (6), constitute a first integral of the supposed equation of the second order. It is evident that (7) is the general type of all such first integrals. Thus the type of the first integrals of (3) would be But any two first integrals included under tliis type and in- dependent of each other would lead us, as is obvious, to tlie same final integral (4), either under its actual or imder an equivalent form. While therefore, viewed as an independent system, the first integrals of a differential equation of the second order are but B.D.E. 2 18 GEOMETRICAL ILLUSTEATIONS. [CH. T. two, it is formally more correct to regard them as infinite in number, but as so related that anj two of them which are independent contain by implication all the rest. Such considerations are easily extended to differential equations of the higher orders. Gej^metrvcal illustraticns, 11. Geometry, by its peculiar conceptions of direction, tangency, and curvature, all developed out of the primary conception of the limit, Art. 1, throws much light on the nature of difterential equations. As the simplest illustration let the equation of a straight line y = ax+h (1) be taken as the complete primitive, a and h being arbitrary constants. Differentiating, we have dy Eliminating a, we find and again differentiating d. '' (^)' y-''%-^ ^^' S=o w- Of these equations, (1), which is free from arbitrary con- stants, is the general differential equation of the second order of a straight line; and (2) and (3), each of which contains one of the original arbitrary constants, are the two differential equations of the first order. ]\Ioreover, each of these dif- ferential equations expresses some general property of the straight line — (2), that its inclination to the axis is uniform ; (3), that any intercept, parallel to the axis of y, between the Ar^T. 11.] GEOMETRICAL ILLUSTEATIONS. 19 straight line and a parallel to it through the origin will be of constant length; (4), that a straight line is nowhere either convex or concave; — and this property, which does not in- volve, in the same definite manner as the others do, the con- siderations of distance and of angular magnitude, is evidently the most absolute of the three. The equation of the circle is {x-af+{y-lf=T' (J), and if we regard a and h as arbitrary constants tlie corre- sponding differential equation of the second order will be ^-^% (6), c£y expressing the property that the radius of curvature is in- variable and equal to r. If we proceed to another differentiation, we find which Is the general differential equation of a circle free from arbitrary constants. And the geometrical property which this equation also expresses is the invariability of the radius of curvature, but the expression is of a more absolute character than that of the previous equation (G). For in that equation we may attribute to r a definite value, and then it ceases to be the differential equation of all circles, and pertains to that particular circle only whose radius is r . The equation (7) admits of no such limitation. Monge has deduced the general differential equation of lines of the second order expressed by the algebraic equation ax' + Ixy + cy' + ex -^-fij = 1 . 20 EXERCISES. [CH. I. It IS '^l'.^V ^ _ 45 ^/ f^/ ^ . 40 r^Y- 0 AxV dx' dx' dx' dx' ■*■ I dxV ~ But here our powers of geometrical interpretation fail, and results such as this can scarcely be otherwise useful than as a registry of integrable forms. From the above examples it will be evident that the higher the order of the differential equation obtained by eli- mination of the determining constants from the equation of a curve, the higher and more absolute is the property which that differential equation expresses. We reserve to a future Chapter the consideration of the genesis of partial differential equations as well as of ordinary differential equations involving more than two variables. EXERCISES. 1. Distinguish the following differential equations accord- ing to species, order, and degree, and take account of any peculiarities dependent upon their coefficients. (1) J-'y=-'. ^ ' cW xdx ^ ' .,. dz dz o? (4) X --. ?/ ^- = — . ^ ^ dx ^ dy y ,^. d^u d\ d\ CH. I.] EXERCISES. 21 _ 2. Explain the tenii ' complete primitive,' and form the differential equations of the first order of which the following are the complete primitives, c being regarded as the arbitrary constant, viz. : (1) 2/ = ca; + V(l + c'). (2) y = {x + c)e". (3) 2, = C6-"°"''+tair'a;-l (4) y={cx + \ogx + lY\ (5) if-2cx-c^ = 0. (6) y = cx + 4,{c). 3. Fomi the differential equations of the second order of which the following are the complete primitives, c and c being regarded as arbitrary constants. (1) y = cQ.o?>mx-\- c ?>mmx. (2) y = G cos [mx + c), c + ex (3) y = x log X X sin mx (4) y = c sin nx + c cos 7ix + 2m 4. State the criterion by which it may be determined whether differential equations are derived from a common primitive. 5. Shew that the differential equations 3,t — = 0, yAl-m-h dx ' ^ I \dx are not derived from a common primitive involving a and h as arbitrary constants. 6. Shew that each of the following pairs of equations, in which ^ stands for ,- , is derived from a common primitive, and determine the primitive : 22 EXERCISES. [CH. I. (2) y — xp = a {if + p), and ?/ — ccp = 5 (1 + a;^^). 7. How many first, second, third, &c. integrals, belong to the general differential equation of lines of the second order given in Art. 11, and how many of each order are inde- pendent ? 8. From the equation {y — hy^ = Am {x — a) assumed as the primitive, deduce 1st the differential equations of the first order involving a and h as their respective arbitrary constants ; 2dly the general functional expression for all differential equa- tions of the first order derivable from the same primitive. 9. Of what primitive involving two arbitrary constants would the functional equation ^{y — 2]jx, p^x) = c, represent all possible differential equations of the first order? 10. How many independent differential equations of all orders are derivable from a given primitive involving x, y, and n arbitrary constants? ( 23 CHAPTER IL ON DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND DEGREE BETWEEN TWO VARIABLES. 1. The differential equations of wliich we sliall treat in this Chapter may be represented under the general form ax J/ and ^ being functions of the variables cc and y. In this mode of representation x is regarded as the inde- pendent variable and y as the dependent variable. We may, however, regard y as the independent and x as the dependent variable, on which supposition the form of the typical equation will be i/^ + jVr=0. For as any primitive equation between x and y enables us theoretically to determine either ^ as a function of x^ or x as a function of y, it is indifferent which of the two variables we suppose independent. It is usual to treat this equation under the form Mix + l^dy = 0, not however from any preference for the theory of infinitesi- mals, but for the sake of symmetry. The order of this Chapter will be the following. As the sohition of the equation, if such exist, must be in the form of a reLation connecting x and y, I shall first establish a prelimi- nary proposition expressing the condition of mutual depend- 24 DIFFERENTIAL EQUATIONS OF THE [CH. II. ence of functions of two variables ; I shall then inquire what kind of relation between x and y is necessarily implied by the existence of a differential equation of the form ax I shall discuss certain cases in which the equation admits readily of finite solution; and I shall lastly deduce its general solution in a series. Prop. i. Let V and v hs exj^Iici'f functions of the tioo vari- alles X and y. Then, if V he expressible as a function of v, the condition dVdv_dVdv^^ dx dy dy dx ^ ^ will he identically satisfied. Conversely^ if this condition he identically satisfied, V will he expressible as a function ofv, 1st. For suppose V—^{v), Then dV^d(j>{v) dv^ dx dv dx ' dV_d4>{v) dv dy dv dy ' Multiplying the first equation by -j- , the second by j- , and subtracting, we have dVch__dV dv^^ dx dy dy dx And this is satisfied identically ; since by the process of elimination the second member vanishes independently both of tlie form of t; as a function of x and y, and of the form of Fas a function of v, 2ndly. Also if the above condition be satisfied identically, Fwill be expressible as a function of v. For whatever func- tions V and v may be of x and y, it will be possible by elimi- ART. 1.] FIRST ORDER AND DEGREE. 25 nating one of the variables x and y to express Fas a function of tlie other variable and of v. Suppose for instance the expression for Fthus obtained to be V=^{x,^). Then dV dx d^(x, ■ dx v) d(j)(x, + dv v) dv ~ Tx' dV Ty- z dcpix, dv v) dv ~ dy- dV dV . Substituting these values of -j- and -7- in the equation (1) we have as the result d<^{x,v) dv^^ dx dy ^''^' But, V being hy hypothesis an explicit function of both variables x and y, -j- is not identically 0. Hence, from (2), dx identically. Therefore (/){a?, ?;), which represents F, does not contain x in its expression j and F reduces simply to a func- tion of V, We have supposed each of the functions F and v to con- tain both the variables x and y. But, whether this be or be not the case, tlie identical satisfying of (1) is the necessary and sufficient condition of the functional dependence of F and V. For suppose either F or v, and for distinction we shall choose V, to be a function of one of the variables only, as x, and F to be a function of v. Then is Falso a function of a?, and as -y- and —r- vanish identically the condition (1) is satis- dy dy ^ ^ ^ tied. 26 DIFFEEEXTIAL EQUATIONS OF THE [CH. II. Conversely, supposing ?? to be a function of a? only, and (1) to be identically satisfied, that equation reduces to whence Fis expressible as a function oi v, 2. The eqiiation M+ N -j- = 0 always involves the existence of a primitive relation between x and y of the form /(^. y) = c, m which c IS an arbitrary constant. Let us first consider what is the immediate signification of the equation ''^^%-' «• We know that if Ace represent any finite increment of x, and A?/ the corresponding finite increment of ?/, — will represent A?/ the limit to which the ratio -~ approaches as Aa? approaches to 0. Let us then first examine the interpretation of the equation J/+.Y^=0 (2). Aic ^ ^ We have x^ = ~ y • The second member of this equation being a function of x and ?/, since M and N are functions of those variables, we may write ^ = '^(^,2/) (3), the form of <^ (a?, y) being known when J/ and N are given. Now if we assign to x any series of values, it is possible to assign a corresponding series of values of ?/, any one of which being fixed arbitrarily all the others will be determined by (3). ART. 2.] FIRST ORDER AND DEGREE. 27 Thus let x^^ cc, , ir^-.-be the series of arbitrary values of a?, and y^ an arbitrary value of y corresponding to x^ as the value of cc, then, representing by Aa^^ the increment of a?^, i.e. the value which being added to x^ converts it into cc^, we have by (3) therefore y, + Ay, = ?/„ + ^ [x^, y,) Ax^. But as Ay^ represents the increment of ?/, coiTesponding to Ax^ as the increment of x^ it is evident that y^ + Ay^ will be the value of y corresponding to cc^ + Ait^^as the value of a;. Eepresenting then this value of y by y^ we shall have 2/1 = ^0 + ^(^0.^0)^^0 = yo+^K.yo) K-^o) W- In like manner we shall find y.=yi + ^[^vy^ K-^J (5)» but, 7/j being already determined by (4), ?/^ is determined, and, continuing the operation, a series of values oi y will be deter- mined, only one of which is arbitrary, while all tlie others are assigned in terms of that arbitrary value and of the known values of x. If, for example, we have the particular equation Ay={x + y) Ax, and assign to x the series of values 0, 1, 2, 3, 4, &c., and at the same time assume that when x is equal to 0, y is equal to 1, we shall have the two following corresponding series of values, viz. x^ = 0, x^ = l, x,^=2, x^= 3, x,= 4, &c, ?/o = 1> 2/1 = 2, ?/, = 5, y, = l'2, y,=27, &c. By assigning a different value to y^, or by assuming arbi- trarily tlie value of some other term of the series y^, y^, y^, c^c. we should find another set of values of those quantities cor- responding to the given values of x. But, in every such set, 28 DIFFEEENTIAL EQUATIONS OF THE [CH. TI. the values of all the terms but one will be determined by a law. Now if the intervals between the successive values of x are diminished, while their number is proportionately increased, each of the corresponding sets of values of x and y will more and more approach to the state of continuous magnitude. And, in the limit, to every conceivable value of x will corre- spond a value of ?/, determined in subjection to a continuous law — to a law however which permits us to assign one of the values of y arbitrarily. The analytical expression of that law will be the solution of the differential equation given. 3. To illustrate the same doctrine geometrically, if a:! and y represent rectangular co-ordinates, any system such as the above would represent a series of points of which the abscissae having been assumed arbitrarily, the corresponding values of y, except one, are determined by a continuous law. In the limit, that series of points would approximate to a curve the species of which as dependent upon the form of its equation would be determined by a law, but an element of which, re- presented by a constant in that equation, would be left arbi- trary, so as to permit us to draw the curve through a given point. The form of the analytical solution thus indicated is /{^,2/)=c (6). The genesis of differential equations of the first order and degree from equations of this description has already been explained in Chap. I. Art. 6. It is evident that, as c is arbi- trary, such a value may be assigned to it as to make a given value of y correspond to a given value of x. If those corre- sponding values are x^^ y^, we have only to assume /k'^o) =c (7), whence c Is determined. But c being once determined, all the values of?/ depend upon those of cc, in obedience to the law expressed by (6). ART. 4.] FIRST ORDER AND DEGREE. 29 Lastly it may be shewn that two distinct complete primi- tives of Mdx + Ndy = 0 cannot exist. For suppose that there are two such primitives w = c, V = C', then by differentiating each du du dy _ r. ^^ I ^^ ^ _ n . dx dy dx ' dx dy dx ' whence, eliminatincr -j-- , ' ^ dx^ du dv du dv _ dx dy dy dx ' ■which shews, by Prop. I, that v is a function of u. The second equation is then equivalent to and this is resolvable by solution into equations of the form each of which is therefore only a repetition of the first sup- posed complete primitive. Certain cases m luliich the equation Mdx + Kdy = 0 admits of finite solution, 4. The equation Mdx + Ndy = 0 can ahvays he solved when the variahles in M and N admit of being sejparated ; i. e, ichen the equation can he reduced to the form Xdx + Ydy = 0 (8), in \61iich X is a function of 's, alone, and Y a function of j alone. To solve the equation in its reduced form (8), it is only necessary to integrate the two terms separately, and to equate the result to an arbitrary constant. Thus the solution will be ^Xdx-\-^Ydy = c (9). 30 DIFFERENTIAL EQUATIONS OF THE [CH. II. On differentiating this result the arbitrary constant c dis- appears, and (8) is reproduced. Thus the solution of the equation xdx + T/d?/ = 0 will be — — ^ = c, A or, since c is arbitrary, The solution of the equation dx ^ ^.y ^Q 1 + ic 1 -\ry will in like manner be log(l + rr)+log(l + ?/)=c; a result which may be simplified in the following manner. We have log(l + a^)(l + ?/) = c; therefore (1 + a?) (1 + ?/) = e^ But a function of an arhitrary constant is itself an arbitrary constant. Hence we may write as the solution {l + x){l+y) = C. Indeed it frequently happens that solutions which present themselves in a transcendental form admit of being reduced to an algebraic form. Thus also the solution of the equation ^^ ^+__^=0 (10) V(i-^=) V(i-/) bemg ^i\\~^x + sin"^^ = c, we shall have, on taking the sine of both members of the equation and replacing sin c by (7, x^^'{l-y■')^y^{l-x')=G (11), which is algebraic. ART. 5.] FIRST ORDER AND DEGREE. 31 5. Different modes of mtegratlon will also give rise to solutions wliicli at first sight appear to be discordant. Tlie discordance however will be only apparent. Thus if we ex- press the equation last solved in the form — dx ^ — dy _ = 0, V(l-a;') V(l-/) and integrate by means of the formula — dx h cos ^x + const., ^f{l-x') we shall have cosT^x + cos"^?/ = (7j and, taking the cosine of both members, xij-^{{l-x'){l-f)} = cosC, (12). The last result may however be reduced to the form x^{l-f)-^7/^{l-x') = smC, (13), which, as sin C^ is arbitrary, agrees with the previous re- sult, (11). The constants C and Cj are seen to be connected by a relation C = sin (7^ , which is independent of the variables x and ?/. And in general the test of the accordance of two solutions of a differential equation, each involving an arbitrary constant, is, that on eliminatwg one of the variables^ the other variahle ivill disappear also, and a relation between the arbitrary con- stants alone i^esidt. Or expressing the solutions in the form we may directly apply the test of equivalence d_Vdv^_dVdv_^^ dx dy dy dx ' resulting from the proposition in Art. 1. 82 DIFFEREXTIAL EQUATIONS [CH. II. 6. It sometimes happens that the variables may be sepa- rated by multiplying or dividing the equation by a factor. Thus the equation xdx ydy ^^ 1 + 2/ 1 + iC becomes on multiplying by (1 + x) (1 + J/), x{l-\-x)dx-y [l+y)dy=^ 0, in which the variables are separated. Integration then gives x^ o? y y^ _ The most general form of equations in which the variables can be separated by the process above mentioned is XJ^dx-\-XJ^dy = ^ (14), in which X^ and X^ are functions of x only, and Y^ and Y^ functions of y only. On dividing the above equation by FjA'25 or, which amounts to the same thing, multiplying it by the factor -y-y , we have '^dx^^^dy^^ (15), in which the variables are separated. Ex. The equation x^J{^^y')dx-\-y sJi^-\- ^)dy = ^ is thus reduced to xdx ydy _ ~^ oT — '-'1 + and lias for its complete integral 7. Sometimes too the variables in the equation Mdx + Xdy = 0 admit of being separated after a preliminary transformation. ART. 7.] OF THE FIRST ORDER AND DEGREE. 33 Ex. 1. If in the equation {x — y^) dx + 2xydij=0^ we assume y = \/{xz), we find ^ _ zdx -\- xdz Substituting these expressions for y and dy in the given equation, we have dx , — + cZ.^ = 0. x Therefore integrating and replacing z hj its value — , X lo": a? + — = c. ° X Ex, 2. {y -x){l + x-)^ dy -n[l + /)^ dx = 0. Assume 07 = tan ^, y = tan cj). Vie find (tan (j) — tan ^) sec 6 sec^0 cZ^ — « sec^<^ sec^^ tZ^ = 0, which reduces to sm{(j>-e)d^-nde = 0. Now let (j)-0 = ylr, then sin '\{rd(j) = ?2c7(/) — WiZ'x/r, nd-^jr therefore dcf) n — sin -v/r ' ndylr - sin the integral in the second member is a known form. whence =\ ^^—r + c: J n — sni Y It will be remarked that the transformations employed in the above examples are not very obvious ones. They would scarcely be suggested by the forms of the difter- ential equations themselves. And in the present state of analysis, it would be impossible to lay down any general di- rection on the subject. There are however certain classes of differential equations in which the nature of the required trans- formation can be determined. Among them a foremost place is due to homogeneous equations. B. D. E. 3 34 HOMOGENEOUS EQUATIONS. [CH. 11. Homogeneous Equations, 8. The diiFerential equation Mdx + Ndy = 0 is said to be homogeneous when M and N are homogeneous functions of X and J/, and are of the same degree. Thus the equation {?/ + V {x^ 4- y'^)]dx — xdy = 0, is a homogeneous equation, M and N being here of the first degree. To integrate a homogeneous equation it suffices to assume y = vx. In the transformed equation the variables x and v Will then admit of separation. Thus in the above example we should find [vx + X \/(l ■\-v^)]dx — x {vdx + xdv) = 0, whence dividing by x V(l + v^) dx — xdv = 0, from which result dx dv log X — log [v + ^/ {I + v^)] = c. Replacing ?; by - , we have log X - log l'^ ^T"^ J " ''' for the complete primitive. As in Art. 5, the above solution admits of a simpler ex- pression. Freed from transcendents and radicals, it gives x' = 2Cy+C% C being an arbitrary constant. To demonstrate the above method generally, let us suppose that M and N are homogeneous functions of x and y of the ART. 8.] HOMOGENEOUS EQUATIONS. 35 n^^ degree. We may then, in accordance with the known type of homogeneous functions, Avrite j/=."<^(|),.v=.y(9, SO that the equation Mdx + Ndy = 0 becomes on substitution and division by the common factor a?", '^($)d^ + ^($)dy=<^ (^•^)- Now assuming y = vx, we have ^ = v, dy = vdx 4- xdv^ X and the above equation becomes {v) dx + '^jr (v) {vdx + xdv) = 0. Or, {(j) {v) + v^lr {v)} dx -V yfr {v) xdv = 0. Therefore /3 + c) dx + [ax + Vy - aa-h'^ + c) dy = 0, whence if a and /S be determined by the conditions aa + Z>/3 = c, a'a + Z^'/8 = c, we shall have the homogeneous equation [ax + ly) dx + (a'ic' -f Vy') dy = 0. Making then y = vx we find dx' {a-\-h'v)dv _ , . x'^^ai- (b + a) V + ^»V" ^^' which is directly integrable. The second transformation gives adx + hdy = dx\ adx + h'dy = dy\ whence determining dx and dy, the proposed equation assumes the homogeneous form (b'x — ay') dx' — {hx — ay) dy' = 0. Both these transformations fail if ah'— ah = 0. But in this case, since h' = ■ — - , the proposed equation may be expressed a in the form {ax + hy-\-c)dx+- (ax + hy + -r]dy = 0, and the variables will be separated if we assume ax + hy = z, and then adopt either z and x or z and y as the new variables. These transformations are linear, and by one of the two the proposed equation is usually solved. 50 6-^^ ,38 LINEAR DIFFEEENTIAL EQUATIONS [CH. II. 10. The linear differential equation of the first order and degree i+^^=^ c^^)' P and Q being functions of x, admits of being solved. When § = 0 the solution is obtained by separating the variables ; and when Q is not equal to 0, a solution may be founded upon that of the previous and simpler case. It must be observed that the linear equation (21), when reduced to the form [Py-Q)dx-Vdy = 0, falls under the general type, Mdx + Ndy — 0. 1st, When § = 0, we have Dividing by ?/, in order to separate the variables ^=-Pdx. y Therefore, log ^ = — 1 Pdx + c, which gives = Ce-f^'-^ (22), C being an arbitrary constant substituted for e'^. It has been already observed that a function of an arbitrary constant is itself an arbitrary constant ; see Art. 6. 2ndly, To solve the linear equation (21) when Q is not equal to 0, let us assign to the solution the general form (22) above obtained, but suppose C to be no longer a constant but a new variable quantity — an unknown function of x, which must be ART. 10.] OF THE FIRST ORDER AND DEGREE. 39 determined in accordance with the new conditions to wliicli the solution must be subject. Substituting then the above expression for ?/ in (21), and observing that, since C is now variable, we have ax dx ax there results dx Hence f-^^^O- . Therefore G = fe-^^'^^ Qdx + c, c being an arbitrary constant. Substituting this generalized value of G in (22), we have finally y = 6--^^' '^'^(^^e^^'^Qdx^c^ (23), the solution required. It will be observed that if (? = 0, the above solution is reduced to the form (22) before obtained. The method of generalizing a solution above exemplified is called the method of the varnatton of parameters, the term parameter, by an extension of its use in the conic sections, being applied to denote the arbitrary constants of the solution of a diiFerential equation. It is only, however, in certain cases that this method is successful. It is always legitimate to endeavour to adapt a solution to wider conditions by a transformation, which, like the above, only introduces a new variable instead of an old one, or a new and adequate system of variables in the room of a former system. But it is not always that the equations thus obtained are, as in the above example, easier of solution than those of which they take the place. 40 LINEAR DIFFERENTIAL EQUATIONS [CH. II. dx x + Ex. 1 . Given $? - -^ = {x-\-\Y Here P^^^. Q = {x + \)\ Hence jP6?a; = -2 log (« + 1), e/'''^^= (aj + 1)-^ ^ef'''' Qdx = \{x + \) dx = ^'^^^^^ + c. Therefore 3/ = (a; + 1)'^ l^^l^ + 4 * Ex.2. Given ^--^ = 6^(0^ + 1)". Here we find jPdx = — n log (a: + 1), ef^'^'^ix + ir, [ef''''Qdx = je'dx = 6\ Therefore 3/ = (a; + 1)" (e'^ + c) . 11. Equations of the form P and Q being functions of x, are reducible to a linear form. For, dividing by ?/", we have Now let 2/'-" = 2, then ^ ' -^ dx dx . ^dy 1 dz whence 2, 5^=r^,te' so that the equation becomes ART. 12.J OF THE FIRST ORDER AND DEGREE. 41 1 dz \— n dx + r^ = Q, or J + (l-,OP. = (l-«)2/) (26), wherein And, continuing thus to repeat the same operation, we obtain ART. 12.] GENERAL SOLUTION BY DEVELOPMENT. 43 a series of equations determining the successive clifFerentiaJ coefficients of?/, in the form g=/»(^.2/) (27), the dependence oi /^{x, y) upon /„_j (a?, ?/), and hence ulti- mately upon/^(;r, ?/), being determined by the general equa- tion f.{^,y)-'^^^-'^^fA-,y) (28). Hence ^l/and JY being given, the expressions for dy d^^i are implicitly given also. Now -r- , -rk •> &c. determine the coefficients of the several dx dx terms after the first in the development of y in ascending powers of x, by Taylor's theorem, or more generally in as- cending powers of x — x^, where x^ is a particular value oi x. Leaving that first term arbitrary, the development is thus seen to be possible, and the result, while constituting the general integral of the given differential equation, shews that that integral involves an arbitrary constant. Actually to obtain the development, let {x) represent the general value of y, and let y^ be the particular value of y corresponding to some particular and definite value, a*o, of the variable x. Then, writing (/> {x) in the form we have, by Taylor's theorem, y = (^„) + " (x,) ^^^3^ + &c. ... (29) . But (.rj = y^. Again, (l>'{x^) is what — — , i. e. ~ , becomes when x = x^. Hence ^' W =/i (^o' 2/o) ^7 (2^)- I^^ ^^^g manner (/>" (o-J is 44 GENERAL SOLUTION BY DEVELOPMENT. [CH. IL 72 what y~ becomes when x — x^^ and is therefore equal to /j(a?o,?/J. Determining thus the successive coefficients of (29), we have finally y = y. +/i (^0. 3/o) (^ - ^^o) +/2 (^0 . 2/o) ^^tI^ + &c.. . . (30), which is the general integral. If we assume x^ = 0, and represent the corresponding value of y by c, we have 3, = o+/(0,c)«+/,(0,c)j^ + &c (31). Should however any of the coefficients in this development become infinite we must revert to the previous form, and give to Xq such a value as will render the coefficients finite, and therefore justify the application of Taylor's theorem. Virtually the integral (30) involves like (31) only one arbi- trary constant. For in applying it we are supposed to give to Xq a definite value, and this being done the corresponding arbitrary value of y^, constitutes the single arbitrary constant of the solution. EXERCISES. 1. Integrate the differential equations : (1) (1 + x) ydx + (1 - ?/) xdy = 0. (2) {y' + xf)dx+{x'-yx')di/ = 0, (3) xy{l + x^)dy-{i-\-y')dx = 0, (4) (1 + f) dx-{y + V(l + /)} (1 + ^^)^ dy = 0. (5) sin X cos ydx — cos x sin ydy = 0. (6) sec'^ic tan ydx + sec^y tan xdy = 0. en. II.] EXERCISES. 45 2. Different processes of solution present the primitive of a differential equation under the following different forms, viz. tan"^ {x + ^) + tan"^ {x - y) = c, y'-x^-^\ = 2Cx, Are these results accordant ? 3. Integrate the homogeneous equations : (1) [y — X) chj + ydx = 0. (2) {2 ^J[xy) -'x]dy^- ydx = 0. (3) xdy — ydx — sj^x^ + y^) dx = 0. (4) (^ — y cos - jdx + x cos - dy = 0. (5) {Sy + Wx) dx + {5y + 7x) dy = 0. 4. Integrate the equations : (1) {2x-y + 1) dx+{2y-x- l) dy = 0. (2) {Sy -7x + 7)dx+ {ly - 3a; + 3) J^ = 0 ; the former as an exact differential equation, the latter by re- duction to a homogeneous form. 5. Explain what is meant by variation of parameters, and, having integrated the equation x -j- — ay =0, deduce by that method the solution of the equation x -^ —ay = x-\-l. 6. Integrate, by the direct application of (23) the linear equations, ^^ dx'^ l-^x''^~ 2x[\-\-x')' (2) x{l-x^)'lL^{:lx^-\)y = ax\ (3) ^ + ;/ ^ a? + \/(l-.y') 46 EXERCISES. [CH. II. , . dy sin 2a; (4) ^ + 2/cosa; = -^. (5) (l + :.')J + y = tan-x. 7. Shew that the solution of the general linear equation -J + Py = Q may be expressed in the form 8. Shew that, <^ {x) being any function of x^ the solution of the linear equation willbe3/ = c6'^("^-^(a;)-l. dti 9. Shew that if in the linear equation -^ -^-Py— Q we represent -f- by ^;, and then, differentiating and eliminating y, form a differential equation between^:? and x^ that equation will also be linear. 10. Integrate the differential equations : (2) 34-a.3 = .+ l. (3) '^^ + 2x, = 2ax'z\ dz (4) T" + ^ ^^^ cc = s" sin 2x. (5) ^^ + y = rios^- ( 47 ) CHAPTER III. EXACT DIFFERENTIAL EQUATIONS OF THE FIRST DEGREE. 1. As the cases considered in the previous Chapter under which the equation Mclx + Xdi/ = 0 is integrable by the sepa- ration of the variables, are but a small number of the cases in which a solution expressible in Unite terms exists, Analysts have engaged in a more fundamental inquiry of which the following are the objects, viz. 1st, To ascertain under what conditions the equation Mdx + Xcly = 0 is derived by immediate differentiation from a primitive of the form f{x, y) — c, and how, when those conditions are satisfied, the primitive may be found. 2ndly, To ascertain whether, when those conditions are not satisfied, it is possible to discover a factor by which the equa- tion Mdx-\-Xdy=^0 being multiplied, its iirst member will become an exact differential. These inquiries will form the subject of this and the follow- ing Chapter. Prop. I. The one necessary and sufficient condition under which the first member of the equation Mdx + Xdy = 0 is an exact differential is dM^dX ,j. dy dx Let it be considered in the first place what is meant by the supposition that 3ldx + Xdy is an exact differential. It is that j\I and X are partial differential coefficients with respect 48 EXACT DIFFERENTIAL EQUATIONS [CH. III. to X and ?/, — that there exists some function F, such that f=-. «. f=* «• Any relation between M and N which we can derive inde- pendently of the form of Ffrom the above equations will be a necessary condition of Mdx + Kdy being an exact diiferential. And conversely, any relation between M and N which suffices to enable us to discover a function V actually satisfying the above equations (2), (3), will be a sufficient condition of Mdx-\-Xdy being an exact differential. And if the same condition should present itself in both cases, it will be both necessary and sufficient. Differentiating (2) with respect to y, and (3) with respect to X, we have d^^d_M d^^dN ,^. dydx dy ' dxdy dx But the first members of these equations being, by a known theorem of the Differential Calculus, equal, we have dM_dN . 'd^-Tx ^^^* This, therefore, is a necessary condition of Mdx + Ndy being an exact differential. It is also, as will next be shewn, a sufficient condition. In the first place the function V, if such exist, must satisfy the equation (2). Integrating this equation relatively to x alone (since the differentiation in -j- is relative to x alone), we have V=\Mdx+C (6), ART. 1.] OF THE FIRST DEGREE. 49 C being a quantity which is constant relatively to x, so that dC -J— = 0. Hence, though C does not vaiy with Xy it may vary with y, and there is nothing to limit the manner of its varia- tion. It is therefore an arbitrary function of?/, and we may write V=JMdx + cl>{y) (7). This is the most general form of V as a function of x and y, which satisfies the equation (2). In the second place Fmust satisfy the equation (3). Sub- stituting in that equation the value of V given in (7), we have dfJIdx ^ d4){y) ^^ dy ' dy Therefore #M=,Y_^f^. dy dy Whence ^[y) ^^{n -^-l^) dy ^. C (8), C being simply an arbitrary constant, since, as the constant of integration with respect to y it cannot contain y, and as part of the expression for ^ {y) it cannot contain x, IN'ow the integration in the second member is theoretically possible (though its expression in finite terms may not be dfVdx possible) if the coefficient of dy, viz. N— -^ — '- , is a function of y only, i. e. if its differential coefficient with respect to x is 0. Expressing this condition, we have dX ddfMdx^^ dx dx dy ^ ^' j^^^ d dfMdx ^ d djMdx dx dy dy dx ^dJI "" dy ' B. D. E. 4 50 EXACT DIFFERENTIAL EQUATIONS [CH. 111. Tims the condition (9) becomes ^y^L^o (10). ax ay This then is a sufficient^ as it has before been shewn to be a necessary condition of Mdx + J^dy being an exact diffe- rential. The substitution in (7) of the value of ^ [y) found in (8) gives F=/j/c?^+|(.V-^)A/+C (11). Finally, supposing still the condition (10) satisfied, the solution of the equation Mdx + Ndy = 0 will be ^Mdx^\{N-^-^^dy=G (12). , 2. The practical rule to which the above investigation leads is the following. To solve the equation Mdx + Ndy = 0 when its first mem- ber is an exact differential, integrate Mdx with, respect to x, regarding y as constant, and adding, instead of an arbitrary constant, an arbitrary function of y, which must afterwards be determined by the condition that the differential coefficient of the sum with respect to y shall be equal to N. Then that sum equated to an arbitrary constant will be the solution required. Elx. 1. Given {x'' - Axy - 2?/) dx + (?/' - 4:xy - 2x^) dy = 0. Here 21= x^ — A^xy — 2y^ and N=y'^ — ^.xy — 2^^, w^hence dy ~ dx~ ^^^ and the first member of the given equation is an exact diffe- rential. ART. 2.] OF TnE FIRST DEGREE. 51 Kow \Mdx = j-2x\j-2fx + (l>{y) (1), the arbitrary function ^ (?/) occupying, according to the rule, the place of the constant of integration. To determine ^(y), we have ^ jl - 2xhj - 22fx + <^ (2/)} =2/= - ixy - 1x\ Whence ^=/, dy -J' Substituting this value in the second member of (1), and equating the result to an arbitrary constant, we have the solution required. . Ex.2. Given ^-.^ + jl _ ^^H/ = 0. Here J/= -— i__ , iV^= - - ^ Hence we find dM_^ -y _dX ^y [x' + 7f)^~ d-^' To obtain the complete integral we will on this occasion employ directly the general form of solution (12). We have Mdx = log [x + sjix" + 2f)], 1^ '^ hidx ' kl' di/J y y\l{x'+ij-)' (12) \og[x + ^|{x" + y')]=c. Ilonoe N— j; \3Idx = 0, so that (12) gives simply 4—3 52 EXACT DIFFERENTIAL EQUATIONS [CH. III. Substituting log C for c, and then freeing tlie equation from logarithmic signs and from radicals, we have f=C'-2Cx. 3. "We may in many cases either dispense with the appli- cation of the criterion (1), or greatly simplify its application, by attending to the two following principles, viz. 1st, If Mdx-\-Ndi/ can be divided into two portions, one of which is manifestly an exact differential, it suf&ces to ascer- tain whether the other is such. 2ndly, If Mdx + Ndy^ or that portion of it which, according to the above principle, it may suffice to examine, can be re- solved into two factors, one of which is manifestly the exact differential of a function of x and ;/, which we will represent by u, then when the other factor is expressible as a function of w, we shall have an expression of the ioxmf[u)du which is necessarily an exact differential. Ex. Given {^ + -;j^^^ dx + ^^y- j:jijz^l dy^O. ^ This equation may be expressed in the form xdx^-ydij + ^ ,f 7 ^ "^N = 0. Now, xdx-\-ydy being an exact differential, it suffices to ex- iidx xclit amine whether the term ,, . g.- is such also. y^J[y'-x) This term may be expressed in the form of the product y ydx — xdy V(/-^') y X the second factor of which is the differential of - . If we y make - = u the product assumes the form -j- r, , which is y ^ ^ V(l-«*) the differential of sin V. ART. 4.] OF THE FIRST DEGREE. 53 The complete primitive is therefore — — ^ + sm ^ - = c. ^ y 4. The converse form of tlie property last noticed is of sufficient importance to be stated as a distinct proposition, namely. Prop. II. If U and « be functions of x and y, and Udu be an exact differential, then U will be a function of u. For Udu = U^r dx-{- JJ -r-dy. dx dy ^ Hence the second member being an exact differential wo have by Prop. i. , . dU du dUdu therefore -7- -j -. — 7- = 0. ai/ ax ax dij Therefore, by the proposition in the first Article of the second Chapter, JJ will be a function of u. EXEECISES. 1. {pi? + Sa^/) dx-\-[\f-\- Zx'y) dy = 0. 2. fn-'^'') Jaj-2^f7y = 0. ^ 2xdx [I Zx^\ , y' \y y > "^ 4. xdx + ydv + ^^y~y-J^ = 0. 5. {l + e^)dx + ^(l- f\ dy = 0. 6. e^ {x" + ?/' + 2x) dx + 2ye'dy = 0. 54 EXERCISES. [CH. III. 7. {71 COS {nx + my) — m sin [mx + ny)] dx + \m cos {rix + my) — n sin [mx + 72?/)} c7?/ = 0. 8. Shew, without applying the criterion, that the follow- ing are exact differentials, viz. _ , xdx + ydy . ydx — xdy ^ 1st, ^-^ + ^—z ~ = 0. {l + x' + f)^ x' + f X 2nclly, ^^^ + y^y -M-^^-^y-\[ydx-xdy). 9. Integrate the above equations. „^ -r . ,1 .• x'^dy — ayx''~'^dx __, , 10. Integrate the equation — 7-2 _ 2a f- ^ Vo? = 0, distinguishing between the different cases which present them- selves according, 1st, as h and c are of the same or of opposite signs ; 2ndl7, as a is equal to, or not equal to, 0. 11. Shew by the criterion that the expression is generally an exact differential, and exhibit the functional , , . , ^3/ , dN lorms which -7— and -7- assume. ay ax ( 55 ) CIIAPTEE IV. ON THE INTEGRATING FACTORS OF THE DIFFERENTIAL EQUATION Mdx-^Xdy=(), 1. The first memLer of the equation Mdx + Ndy = 0 not being necessarily an exact differential, analysts have sought to render it such by multiplying the equation by a properly determined factor. Thus the first member of the equation (1 + y-) dx + xydy = 0 is not an exact differential, since it does not satisfy the con- dition -i- = -^ , but it becomes an exact differential if the ay ax equation be multiplied by 2:c, and its integration, which then becomes possible, leads to the primitive equation The multiplier 2x is termed an integrating factor. We propose in this Chapter to demonstrate that integrating factors of the equation Mdx + Ndy = 0 always exist, to in- vestigate some of their properties and relations, and to shew how in certain cases integrating fiictors may be discovered. To complete this subject we shall, in the next following Chapter, investigate a partial differential equation, upon the solution of which their general determination depends, and shall examine some of the conditions under which the solu- tion of that equation is possible. 56 ON THE INTEGRATING FACTORS. [CH. IV. 2. To every differential equation of tlie form Mdx + Nd7/ = 0, pertains an infinite number of integrating factors, all of which are included under a single functional expression. It has been shewn, Chap. ii. Art. 2, that the above equa- tion always involves the existence of a complete primitive of the form '^{^,y)=G (1). Differentiating the last equation, we have dyfr {x, y) ^ d-ylr {x, y) dy ^^ ,^.^ dx dy dx The value of -—- determined as a function of x and ?/ from dx -^ this equation must be the same as the value of — furnished by the given differential equation expressed in the form dx Hence eliminating -j- between these equations we have ds^r (x, y) dyjr (x, y) dx dy 3r~~ N ;3). Let IX be the value of each of these ratios, then ^^ (^> y) _ „ 7,7 ^±S^ _„xr dx "^^'^' dy~'~^^' As fiM and fxN are therefore the partial differential co- efficients with respect to x and y of the same function -v/r {x, y), the expression fjuMdx -^ fiNdy will be an exact differential. Thus Mdx-\-Ndy is always susceptible of being made an exact differential by a factor /x. ART. 3.] ON THE INTEGRATING FACTORS. 57 3. The form of the complete primitive is however without gain or loss of generality susceptible of variation. Thus the primitive x'^ (1 + ?/^) = c, Art. 1, might, w^ithout becoming more or less general, be presented in the forms sin {x' (1 +y)} =c„ log K(l + 2/^)1 = c„ or in the functional form/{a7^(l +?/^)} = c, where c, c^, c^ are arbitrary constants. And generally a complete primitive ex- pressed in the form V= c may be expressed also in the form f{y) = c, f{V) denoting any function of V. These variations in the form of the complete primitive imply corresponding variations in the form of the integrating factor, a special deter- mination of which has already been given, Art. 1. To investigate the general form under which all such special determinations are included, let us suppose /x to be a particular integrating facto"r of Mdx + Ndy^ and let fiMdx + fiKdy be the exact difterential of a function -^ [x, y). Then representing for the present ^fr {x, y) by v, we have jjiMdx + ^Ndy = dv. Multiply this equation by /(i;), an arbitrary function of v ; sucli being, by Art. 4, Chap. III., the general form of a factor which will render the second member an exact differential. We have fif{v) [Mdx + Xdy) =f{v) dv. Now the second member of this equation being an exact dif- ferential the first is so also. As moreover the first member of the above equation can only become an exact differential simultaneously with the second, the factor /jif(v) is the general form of a factor which renders 2Idx + Ndy an exact differential. We may express the above result in the following theorem. If /jl he an integrating factor of the equation Mdx + Xdy = 0, and ifv = c he the complete primitive ohtained hy multij^Jying the equation hy that factor and integrating, then /jf{v) will he the typical form of all the integrating factors of the equation. Furthermore, /(v) being an arhitrary function of v, the num- ber of such factors is infinite. 58 ON THE INTEGRATIXa FACTORS. [CH. IV. Ex. The equation {x'y - 2?/^) dx + [ifx - 2x^) dy = 0, becomes integrate on multiplying it by the factor [— ) , the actual solution thus obtained being y X Hence the general form of the integrating factor of the equa- tion is fy'^Kf^xV' 4. From the typical form of the integrating factor of the equation Mdx + Ndy = 0, it follows that if we know two par- ticular integrating factors of the equation, the solution may be infen-ed without integration. For fjb being one of the factors given, the other must be of the form /JLf{v). If we determine their ratio by division and equate the result to an arbitrary constant we shall have f{v)=c, which, from what has been said in the preceding Ai'ticle, is a form of the complete primitive. 5. It has been observed, Art. 1, that the discovery of an integrating factor of the differential equation Jfdx + Ndy = 0 generally depends on the solution of another differential equa- tion, but there are some cases in which it presents itself on in- spection. The equation ixy'^ -\-y)dx — xdy = 0, becomes integrable on being multiplied by the factor -^ , and this factor is at once suggested if we place the equation in the form y'^xdx + ydx — xdy = 0. ART. 6.] SPECIAL DETERMINATIONS. 59 We could thus, also by inspection, assign the integrating factors of any equation of the form T^dx + 0 {x) {ydx — xdy) = 0, and many other forms will readily suggest themselves. The following analysis will however lead to results of greater generality and importance. Special Determinations of Integrating Factors, 6. Whatever may be the constitution of the functions M and N ^YQ have identically But f+f = .log(.,),5-f = .log(f). Hence, Mdx+Ndy = i {{2Ix + Ny) d log xy + [2Ix - Xy) d log 4 (1) • The functions Mx + Ny and 2Ix — Ny appear in the second member of this equation as the coefficients of exact differ- entials. And upon the nature and relations of these functions the inquiry will now depend. Whatever may be the constitution of il/and iV^ some one, and oiily one, of the following cases will present itself. Either the functions Mx + Ny and 2Ix — Ny will be both identically equal to 0, or one of them will be so and not the other, or neither of them will be identically equal to 0. These cases we will separately consider. 1st. The case of Mx + Ny and Mx — Ny being both iden- tically equal to 0 may be dismissed, as it would involve the supposition that M and N are each identically equal to 0. 60 SPECIAL DETERMINATIONS [CH. lY. 2ndly. Suppose that one of the functions Mx + N?/ and Mx — Nt/ is identically equal to 0 and not the other, and first let Mx + Ny be identically equal to 0, then (1) becomes 3Idx + Nd7/ = i{Mx-N7/)dhg-; iJ whence dividing by Mx — Ny, Mdx + Ndy . j, x .^. Mx-Ny -i'^^'-y ^'^' !N"ow the second member being an exact differential the first member is also one. In this case then 2Idx + Ndy is made an exact differential by the factor -^ ^ . By parallel reasoning it follows that if Mx — Ny is identically equal to 0 and not Mx + Ny^ an integrating factor of Mdx + Ndy will be— i— Mx + Ny' And thus we are led to the following theorem. Theorem. If one only of the functions Mx + Ny and Mx — Ny is identicaUy equal to 0, the reci'procal of the other function will he an integrating factor of the equation Mdx + Ndy = 0. 3rdly. Let neither of the functions 2Ix + Ny and Mx — Ny be identically equal to 0. Then first dividing the funda- mental equation (1) by Mx + Ny, we have Mdx + Ndy - ,, 1 Mx — Ny -,. x .„, ■-^, ^^ = i^lof?a?v + i-r7 T# ilog-... (3). . Mx^-lSy 2 ^ ^ ^ Mx-\-]Sy ^y ^ ^ Now, by Art. 3, Chap. III. the second member of the above equation becomes an exact differential (its first term beino: already such) if ^^ ^ is a function of log - ; there- ^ -^ ^ Mx + Ny y X fore if it is a function of - : therefore if it is a homogeneous y function of x and y of the degree 0, for the typical form of ART. 6.] OF INTEGRATING FACTORS. 61 such a function is (-); tlierefore, finally, if J/ and J\" are homogeneous functions of x and ?/ of a common degree. For let M and N be homogeneous and of the n^^ degree. Then Mx — Ny and Mx + Ny are each of the degree 7i+ 1, and Vt ^ is of the deffree 0. Thus ilf and N beinoj homoireneous Mx-\-^y ^ o o functions of the n^^ degree, the second member, and therefore the first member of (3), is an exact differential. From this conclusion, combined with the previous one, we arrive at the following theorem. Theorem. The equation Mdx + Ndy = 0 ivlien homogeneous is made integrahle hy the factor — :r^ , unless Mx + Ny is identically equal to 0, in which case -ry ^^ is cm integrating ■ t/ factor. Always then the homogeneous equation Mdx + ^dy = 0 is made integrable either by the factor -^tt tt- » oi' T^Y the ^ -^ 2Ix + Ny' -^ factor -Y7 TT" • 3Ix - Ny In the second place, dividing the fundamental equation (l) by Mx — Xy, we have Mdx + Nchi , fMx -vNii T, ^ , x —Tr T^=\\-Tr V^ " iO^XV + d loST - 2Ix — ^y ^ \Mx -Ny ° ^^ ^ y of which the second member, and therefore also the first member, becomes an exact differential if 4-?? zr- is a func- Mx — Ny tion of log xy ; therefore if it is a function of xy ; therefore, finally, if M and N are of the respective forms M = F^{xy)y, N = F^{xy)x; 62 SPECIAL DETERMINATIONS [CH. IV. since this supposition would give 3fx + Ny _ F^{xy) + F^{xy) Mx-Ny-F^{xy)-F^[xij)' of which the second member is a function of the product xy. Hence the following theorem. Theorem. The equation Mdx + Ndy = 0 is made integralle ly the factor ^i v~ ' ^^^^^ -^^ ^''^^ ^ ^^^ ^f *^^^ respectice forms M=.F^[xy)y, N = F,^[xy)x, unless ' Mx — Ny is identically equal to 0, in which case 1 is an integrating factor. Mx + Ny Or the theorem might be thus expressed. Tlie equation is made integralle hy the factor 1 ^y[FA^y)-FAxy)V unless we have identically F^ (xy) — F^ (xy) = 0, in which case 1 ^y{K{i^y)+F^{xy)] is an integrating factor. We may, however, remark that, in the particular case in which F^{xy) — F^{xy) = 0, no factor is needed, as the dif- ferential equation may then be expressed in the form F^ (xy) {ydx + xdy) = 0, the first member being manifestly an exact differential. 7. The results of tlie above investigation may be summed up as follows. If either of the functions Mx + Ny, Mx — Ny is identically equal to 0, the reciprocal of the other function is an integrating ART. 7.] OF INTEGRATING FACTORS. 63 factor of Mdx + Ndy = 0 ; hut if neither of these functions is equal to 0, then -. ^ ^ is an integrating factor for the etiuation tvhen homogeneous, and -^t ^ an integrating factor of the equation when suscejptihle of exj^ression in the form F^ {xy) ydx + F^ {xy) xdy = 0. Ex. 1. Given x\lx + (3x'?/ + 2/) dy=^0. This is a liomogeneoiis equation, and its integrating factor according to the rule above given will be 1 Thus we have, as an exact differential equation, x'dx {^x'y-\-2f)dy _ x\+'6xY + 2y''^ x'+^xSf->r2y' ^ ^^^• Eeferring then to Art. 2, Chap. ill. we have r r x?dx j j X -\- dxy + 2y dx 2x X jW+2/ x' + y Differentiating this expression with respect to y, and com- paring the result with the corresponding term in (]), y/e find 7 ' = 0, whence (?/) = const., and we have or x'+2y'=C^{x'-\-y') for the integral required. 64 SPECIAL deter:\[inations [ch. IV. Ex. 2. Given [y + xi/^) dx+ {x — yx^) dy = 0. This equation may be expressed in the form (1 4- xy) ydx + (1 — xy) xdy = 0. Hence its integrating factor, as given by the rule, will be 1 ^ 1 Mx — Ny (1 + xy) xy — (1— xy) xy 1 "" 2afy' ' Eejecting the constant J, we have, on multiplying the given equation by ^, , Hence xy xy^ "^ Now ISldy = -^ — -, Hence the complementary function ^ [y) will be - log y. Thus we have \ogx-\ogy---^G for the integral required. Ex. 3. Given {x^y"^ + xy^) dx — [x^y + xHf) dy = 0. If we treat this as a homogeneous equation regardless of the implied conditions, we find Mx+Ny 0* The rule however shews that when Mx-\-Ky is, as in the ART. 9.] OF INTEGRATING FACTORS. 65 above example, identically equal to 0, -. , _ .^ represents an integrating factor, which in the above case will be 1 The equation is thus reduced to dx dy ^ y whence we find y = cx as the complete integral. 8. From the theorems of the preceding article others of greater generality may be deduced by transformation. Thus, since the equation F^ [xy) ydx + F^ [xy) xdy = 0 is made inte- grable by the factor 7-^77 — \ _ t^ 1 — \\ -> i* follows that the xy\r^\xy) r^[xy)\ equation jPj [iLiv) vdu + jPg {uv) udv = 0 is made inteOTable by the factor — t^ti — ^ ttt — d » ^ ^^^ ^ ^ uv [F^ {uv) - F^ {uv) j ' V being any functions of x and y. Hence expressing du in the form -j-dx+ -j-dyj and dv in the form -T-dx-\--^ dy^ we see that the equation is made inteo^rable by the factor ttt^ — r ^=r-^ — rr ? "what- ° ^ uv[F^[uv)-F^{uv)\' ever functions of x and y are represented by u and v. And, on giving particular forms to these functions, particular con- ditions of integration of the equation Mdx + Ndy = 0 present tlicmselves. 9. An integrating factor for homogeneous equations may also be found by the following method, due to Professor Stokes, who first pointed out the necessity of taking account of the B. D. E. 5 66 HOMOGENEOUS EQUATIONS. [CH. IV. case in which Mx + ^y is identically equal to 0. ( Cambridge Mathematical Journal, Vol. IV. p. 241. First Series.) Suppose M and N to be homogeneous functions of x and y of the degree n. Then we may write ilf=a;"(^(v), iY=a;"A;r(v) (l), where v stands for ^ . X Hence Mdx + Kdy = x''<^[v)dx-^x'''>^{y)dy (2). But y = XV, therefore dy = xdv + vdx. Substituting this value of dy in the second member, we have Mdx + Ndy^x"" [j>{v) + v^\r {v)] dx + aj"-''^ (v) Jv...(3). Two cases here present themselves. First, the constitution of the functions ^ [v) and -^^ {v) may be such that ^ (v) + i-v|r (v) may be identically equal to 0. This will happen if 2Ix + Xy is identically equal to 0, since V(l) 3Ix + Ky=:x''^'{^{v)+vf{v)] (4). In this case the equation (3) reduces itself to Mdx + Ndy = x""^' a/t iy) dv, Mdx + Ndy , / V 7 or -^^^r-^='^[^)dv. Now the second member being an exact differential the first is so also, and Mdx H- Ndy is therefore made integrable by the factor -—-TT-. a; ^ Secondly, the constitution of ^{v) and ^/^(v) may be such that '(x) dy dx, {x) N ' dM dN dx^°^' 4>{x) = dy dx Xow if the second memher of this equation is a function of x the equation is integrable, and we have dM dN ■I- log ^ {x) =] ^ ^j dx. "WTience f^ = eJ ^ (2). 'We have seen that the hypothesis assumed as the basis of the above solution, viz. that the integrating factor ;/, is a function of x only, is legitimate when the constitution of the functions 21 and N is such that the expression /dM dN\ ^ ^ \dy dx)' is a function of x only. In this case (2) enables us to deter- mine the value of /x. In like manner the condition under which //, is a function of?/ only, is dN_dM y =2, function of y only (3), ART. 2.] OF INTEGRATING FACTORS. 71 and the value of fx^ on this hypothesis, is dx - dy ^ = ei'--~'' (4). Ex. Let us inquire whether the equation (3ic'4-6icy + 3/)c?a;+(2a;' + 3xy)^i/ = 0 (5) admits of an integrating factor which is a function of x only. Making M= Zx^ + 6xy + 3?/', N= 2x'' + Sxi/, we find dy dx 6a; + 6,y - {Ix 4- 3y) _ 1^ 2Y ~ 2x' + Sa^y ~ a; ' and this result being a function of x alone, the determination of /L6 as a function of x alone is seen to be possible. From (2) we now find Cdx fJi = 6^' = CX, C being an arbitrary constant. Now multiplying (5) by Cx, we have C{{Sx^ + (jx'y + Sxy') dx + {2x' + Sx^y) dy] = 0. The first member of this equation remains a complete differ- ential whatever value we assign to G, If we make (7=1, and integrate, we find the integral sought. The student may obtain also the same result by solving (5) as a homogeneous equation. The linear differential equation of the first order i^^y-Q=' («)' P and Q being functions of x, may be solved by the above method. For, reducing it to the form (Py-Q)dx + dy=0 (7), 72 GENERAL DETERMINATION [CH. Y. we have M=^Py — Q, N= 1, whence dM_dN dy dx _ lY ^' which being a function of x we find from (2) Multiplying (7) by the factor thus determined, we have ef'"" [Py - Q) dx + e/^"^ dy = 0, the first member of which is now the exact differential of the function ef''^y-U''^Qdx, Equating this expression to an arbitrary constant c, we find y = e-f''^{c+^ef''^Qdx] (8), which agrees with the result of Art. 10, Chap. ii. 3. Let it he required to determine the conditions under which the equation Mdx + Ndy = 0, can he made integrahle hy a factor ix which is a function of the product xy. Eepresenting xy by v and making fjL = (j){v), the partial dif- ferential equation (1) becomes ATJ'/ \dv -nrj'r \ dv fdM dN\ , , , whence, since "r = 2/, ^ = cc, we find dM_dN (-) let us represent - by v, and then assuming fM = (j){v), and observing that dv _ — y dv _1 dx ic^ ^ dy x^ ART. 4.] OP INTEGRATING FACTORS. 75 the partial differential equation (1) becomes -^^-Wj-J/fwU(f-f)^W (12), dy _dM\ dy) ,(dN dl 6' (v) \dx dy whence -j-j-r = — tt — --rr — • (j) {v) MX + i\y Thus the condition sought is that the second member of the above equation should be a function of v, i. e. of - . X And the corresponding value of yu. is I \dx dy). But since every function of - is homogeneous and of the X degree 0, with reference to the variables x and y, we may express the above results in the following theorem. In order that the equation Mdx + Ndy = 0 may he made integrahle hy a factor jx which is a homogeneous f unction ofx and y of the degree 0, it is necessary and sufficient that the function \dx dy J . Mx + Ny ^^^^ should he also homogeneous and of the degree 0. This con- dition heing satisfied, the value of fx icill he fjL = e-^'^''"' (U), lohere v stands for - , and f{v) is what the function (13) is X reduced to hy this transformation. The above investigation fails when the constitution of the functions M and N is such that we have identically Mx + Ny = 0. An integrating factor for this case has already been found in the preceding Chapter. 7& GENERAL DETERMINATION [CH. V. We proceed to notice some of the consequences of the above theorem. It is evident that the condition which it involves will be satisfied when J/ and Nslyq homogeneous functions of x and ?/. For, supposing them to be homogeneous and of the n^^^ degree, the numerator and denominator of the fraction (13) will each be of the {71 + ly^ degree, and the fraction itself therefore of the degree 0, the condition required. It is not however by homogeneous equations only that this condition is satisfied, and it is sometimes worth while to inquire into its applicability in other cases. Thus for the equation - + sec^ 1 dx — — „ du = 0 ,2j x) f -^ we should find the integrating factor cos - . X 6. It I's required to investigate the conditions under which the equation Mdx + Ndy = 0 can he made integrahle by a fac- tor fjL, which is a homogeneous fanctio7i of the degree n. Assuming fjL = x'"(f>[-] , the partial differential equation (1) becomes N \ nx =(f-f)-*ffi- Dividing by x""'^ and transposing, we get Mx + Ng whence ART. 5.] OF INTEGRATING FACTORS. 77 Let - = ?;, and suppose the second member to assume the form/(v) ; then, multiplying both sides by dv and integrating, we have Hence ^i = x""^ {v) = x'^e-^'^'^'^ Thus we arrive at the following theorem. Theorem. In order that the equation Mdx + Ndy = 0 may he made integrahle hy a factor /x, which is a homogeneous fane- j| tion of X and y of the li}^^ degree^ it is necessary^ and it suffices, ' that on making y = vx the function .[dN dM\ xH-j j-] + nNx (1^) Mx + Ky sjwuld assume the form f{v). This condition heing satisfied, the expression for /jl ivill he ^ = aj"e/-^(^)'^' (16). It will be noted that the condition that (15) shall be a function of t*, is the same as the condition that it shall be a homogeneous function of x and y of the degree 0. The theorem fails when Mx + Ny = 0, a case already con- sidered. Ex. 1. Required to determine whether the equation [2x' + ^^x'y + ;y" - ^f) dx + (2^^ + Zxy"" + x" - x') dy = 0 admits of an integrating factor which is a homogeneous func- tion of X and y. 78 GENERAL DETERMINATION [CH. V Here if = 2aj' + Zx^y ^f-y\ N= 2y' + Sxy'+ x'- x\ Hence, on substitution, \dx dy J Mx + Ny - (n + 6) a;' + i^n + 6) ccy + Inxif^ [n + 2) a;^- 2a;'y " 2^3^ + 2ic'?/ + 2a?3/' 4- 2^' + cc''2/ + xy"- "We are now to inquire whether there exists any value of n which reduces the second member of the above equation to a homogeneous function of x and y of the degree 0. That member may be expressed in the form -X {n + 6) a;' - (3?i + 6) xy^ - 2?i/- (n + 2) x'' + 2xy x + y^ 2x'+2y' + xy and it is now plain that if any value of 71 will answer the required condition, it must be one which will make the terms containing xy^ and x^ in the numerator of the second factor vanish. Making then ?i = — 2, we have — X 4.x^ + 4?/^ + 2xy _--2x x + y 2x^ ■\-2y^ -{- xy x-\-y - -^ "1 + 2;* ■/;:?■ ■ <^ • Hence \j. = x "e' x^i^^v)' c {X + yf ART. 5.] OF INTEGRATING FACTORS. 79 Multiplying the given equation by this factor and integrating, we find as the primitive equation x-^y In the case of homogeneous equations the condition in- volved in the general theorem will be satisfied independently of the value (f n, the particular case in which Mx + Ny = 0 excepted. It follows hence that with this exception we can find an integrating factor of any proposed degree for the homogeneous equation Mdx + Ndy = 0. Ex. 2. Kequired two integrating factors of the respective degrees 0 and 1 for the equation {^x + 2y) dx + xdy = 0. First making il/= Sx + 2y, N= x, and w = 0, we have ,(dN dM\ \ax ay J _ ~^ Mx + Ny 3 (, x-^y)' /W = -1 "3(l + t;)' ^ ^ ^fmdv = c(?; + l)-^ = f X \h = c — -— . \x-\-yJ Secondly, making M= \dx 3a? + 2y, N= dM\ ^ , . X, ?i = l, we :0. have Mx + Ny Hence / {v) = 0, Thus replacing each of the constants c and c by unity, the integi'ating factors in question are ( Y and x. 80 GENERAL DETERMINATION [CH. V. Multiplying by the second factor x and integrating, we 6nd ix? + y?y = G for the primitive. Again, if in illustration of the remark of Art. 4, Chap. IV., we equate to an arbitrary constant the ratio of the second factor to the first, we have Q^[x-\-y^— constant, which being equivalent to ^ (x + y) = constant, agrees with the previous solution. Let us next examine the general results to which the theorem leads, when M and N are homogeneous and of the 7?t*^' degree. The general forms of If.and N will be on putting v for - , Hence, observing that ^ = mx'^-'^l, {v) - x'^-'yf' (v), we have on substituting in the expression for f[v), and dividing numerator and denominator of the result by a?"'"^^, 4^(,,\ _ (^ + ^) -f {v) - vjr' (v) - ' {v) ^^^^~ (P{v)+vylr{v) ^ ^* If we make ??, the value of which may be chosen at plea- sure, equal to — w — 1, we have ^ ^ ^ ^ {y) + v^ {v) Multiplying by dv and integrating. /. f[v) dv = - log {^ [v) + vf {v)]. ART. 5.] OF INTEGRATING FACTORS. 81 Hence, ^n^/^'^)^" c C Mx + Ni/ :i8). And here again it results that the homogeneous equation Mdx + Ndj/ = 0, may be made integrable by the factor -ry ^, except in the particular case in which the con- stitution of ilf and iVis such as to make Mx-\-Ny=0. More- over this theorem is seen to be only a particular consequence of the general theory of the integrating factors of homogeneous equations. Resuming (17) which we may write in the form ■f( \ — (^ + ^^ + 1) '^ h^)—{^ (^0 + vylr'Jv)+_(p'_{v) } we have by the substitution of which, combined with the previous re- duction, the general value of fju becomes -,jn+n+l Jm+n+l)J-- , , , , yl/{v)dv ^= — Aiic+jry — (^^)' which is the general expression for an integrating factor of the ji^^ degree, supposing ii not equal to —m—\. If we noAv equate to an arbitrary constant the ratio borne by the last value of /u, to the previous one (18), we have r ■, and the condition of the theorem is satisfied. 7. All the applications which we have hitherto made of the partial differencial equation (1) are of one kind. The general problem which they exemplify is the following. Under what condition does the equation Jldx + Xdy = 0 admit of being made integrable by a factor of the form (v) where v is a known and definite function of x and y ? Let us examine the general form of its solution. !2). The condition sought then is tliat the second member of this equation should be a function of v, Representing that func- tion by/(f) the corresponding value of yu, is M = e/'"* (23). Any special case may be treated either independently as in the previous examples, or by directly referring it to the above general form. 6—2 On substituting

lj\ dv (j) (v) + vyjr (v) (p {v) + v-^ {v) '*" '^* Now the reducibility of an equation of this form to a linear form has been established in Chap. ii. Art. 11. Under the general form (24) are virtually included some remarkable equations which have been made the subjects of distinct investigations. Thus Jacobi has, by an analysis of a very peculiar character, solved the differential equation. (Crelle's tfournal^ Vol. xxiv.) {A + A'x + A'y) [xdy - ydx) - {B+ B'x + B"y) dy ^{C-^G'x+ G"y)dx=^0 (26). If, however, we assume in that equation x=^+a, y = 7] + P, we can, by a proper determination of the constants a and ^, reduce it to the form (^1 + «'^) [^^V - ¥^) - (^? + ^V) d7) + (c? + c'77) d^ = 0, which fiills under (24). On effecting the substitution in ques- tion the equations for determining a and (^ will be found to be a[A+ A' a + Jl'^) -{B + B'a + B" (3) = 0, - ^ (^ + ^'a + A!' 13) + 0 + (^'a + C"/3 = 0. The most convenient mode of solving these equations is to write tliem in the symmetrical form B+B'a + B"/3 a+C'a-\-C"l3 , , .. ..^ = o = A + Aql + A /3, then, equating each of these expressions to X, we find A-\-\-A'a+A"l3 = 0, B-\-{B'-\) a+^"/3 = 0, C+C'a + {C"-\)i3=0, 86 GENERAL DETERMINATION [CH. V. from \y1iIc1i eliminating a and ^ we liave the cubic equation {A - \) {B - \) ( G" - X) - B" C [A-\)- A" C [B'- X) '-A'B{C"-\) + A'B"C+A"BC'=0 (27). If a value of \ be found from this equation, any two equa- tions of the preceding system will give a and /3. 9. The present chapter would be incomplete without some notice of a method which was largely employed by Euler. That method consisted in assuming //, to be a function definite in form as respects the variable y, but involving un- known functions of x as the coefficients of the several powers of ?/. After the substitution of this form of /jl in the partial differen- tial equation (1), the result is arranged according to the powers of y, and the coefficients of those powers separately equated to 0. This gives a series of simultaneous differential equations for the determination of the unknown functions of x. But for the success of the method it is necessary that the primary assumption for jul should have been chosen with some special fitness to the object proposed. The following is an example. Required the conditions imder which the equation Fydx+{i/+Q)d7/=0 admits of being made integrable by a factor of the form 1 P, 5, B and S being functions of x. In the partial differential equation (1), making 1 M=:Py, N=y-hQ, f,= f^lif + JSy' ART. 9.] OF INTEGRATINa FACTORS. 87 clearing tlie result of fractions and arranging it according to the powers of y, we have *(«2-«S)^-» M- Whence, equating separately to 0 the coefficients of the dif- ferent powers of y, we have the ternary system --S-S- (-). ^^■^^S-<^f-f=« (^«). ^i'-«S- (-)• The last equation gives S=cQ,c being an arbitrary constant. Substituting this value of S in the equation obtained by eliminating F from the first two equations of the system, we find (2c -B)dQ + 2 QdR = BdE, or, regarding therein E as the independent and Q as the de- pendent variable, a linear equation of which the solution is ' Q = E-c-hc {E-2c)\ Hence we have S^c{E-c)+cc {E-2c)% and from the substitution of the value of Q in the first equa- tion of the ternary system, P=-c'(iJ-2c)g. 88 INTEGRATING FACTORS. [CH. V. These values of S, Q, and P, in which R is arbitrary, re- duce the given differential equation to the form {R-c + c {R-2cy-\- y] dy - cy {R-2c) dR = 0... (32) , and present its integrating factor in the form 1_ - y' + Rf + [c {R-c)+ cc {R - 2cY} y ' R being an arbitrary function of x. For other examples the student is referred to Lacroix (Traite dii Calcul Biff, et du Calcul Int. Vol. II. Chap. IV.) The results of this method are usually of a very complex character, while their generality is limited by the restrictions which must be imposed in order to render the system of reducing equations solvable. Thus Euler's equation above considered is virtually only a limited case of the general equation (21). If we assume y -\- c = s, R — 2c = t, it becomes (s +t)ds + cctdt + c't {ids - sdt), which evidently falls under that equation. EXERCISES. 1. The following equations admit of integrating factors of the form {x) , viz. (1) {x''-i-y^ + 2x)dx + 2ydy=0. (2) {x^+y^)dx-2xydy=0. Determine these factors and integrate the equations. 2. The equation 2xy dx + (y^ — Sx^) dy = 0, has an inte- grating factor which is a function of y. Determine it, and integrate the equation. 3. Find those integrating factors of the equation ydx -f {2y ^x) dy = 0 CH. Y.] EXERCISES. 89- wliicli are homogeneous functions of x and y of the respective degrees 0 and — 2, and from the consideration of those factors deduce the complete primitive of the equation. 4. For each of the following equations examine whether there exists an integrating factor /x satisfying the particular condition specified, and if so determine the factor, and integrate the equation. (1) y {p?-\- y^) dx + X {xdy — ydx) = 0, ^a a liomogeneous function of the degree — 3. (2) (y^ + axy"^) dy - ay^dx + {x + y) {xdy — ydx) = 0, fi as in the previous example. !) {y — x)dy + ydx — xd l-j = 0, /ll homogeneous of the degree — 1. (4) (x^ + y'^+l) dx — 2xydy = 0, /^ a function of y' ^ x^. (5) ( ?/ - 3^y - 2x^) dx + (2?/' + 3xY -x)dy=0, fia func- tion of x^ + y, (6) {x''+x'^y+2xy-y^-y^) dx + (/ + xy"-+2xy -x'-x^)dy = 0, fi a function of the product (1 -\-x) {l+y). 0) {^y^ — x) dx + (2?/^ — Qxy) dy = 0, /x a function of x + y'. 5. The equation y {of + y") dx-\-x [xdy — ydx) =0 has an integrating factor of the form e^ (/> {x^ + y"^). Determine it, and, from the comparison of the result with that of (1) Ex. 4, deduce the complete primitive. 6. The linear equation -jL-\- Py = Q having an integrating factor of the form e/^'*^, deduce a corresponding expression for an integrating factor of the equation 7. Prove that the equation dii 2 dP -r»o 90 EXERCISES. [CH. V. where P is any function of x, has an integrating factor of the form -. j^, Lacroix, Tom. ii. p. 278. 8. Deduce a similar expression for an integrating factor of the equation -^+y'^ + —--^J>^=o. lb, 9. Investigate the conditions under which the equation where P and Q are functions of Xj can be made integrahle by a factor of the form-, ^, ,,„, and determine the form of/(aj). ( 91 ) CHAPTER YI. OF SOME EEMARKABLE EQUATIONS OF THE FIRST ORDER AND DEGREE. 1. There are certain differential equations of the first order and degree, to which, in addition to their intrinsic claims upon our notice, some degree of historical interest belongs. Among such, a prominent place is due to two equations which, having been first discussed by the Italian mathema- tician Pticcati and by Euler respectively, have from this circumstance derived their names. To these equations, and to some other allied forms, the present Chapter will be devoted. Eiccati's equation is usually expressed in the form 2+^'''=-"' «• But as both it and some other equations closely related to it and possessing a distinct interest, may, either immediately or after a slight reduction, be referred to the more general equation xf^-ay^hf=cx'^ (-)' the discussion of which happens to be much more easy than that of the special equations which are included under it, we shall consider this equation first. To reduce Riccati's equation under the general form (2), it suffices to assume w = - . We find, as the result of this X substitution in (1), ^%ry + h-'=^ox-*'- (3), which is seen to be a particular case of (2). 92 OF SOME REMAEKABLE EQUATIONS OF [CH. VI. Of the equation x y-—ciy-\-^y^— ca;". 2. The discussion upon whicli we are entering may be divided into two parts. First, we shall shew that the equa- tion is solvable when n — 2a. Secondly, we shall establish a series of transformations by which a corresponding series of other cases may be reduced to the above. 3. First. The equation x-j — ay -{-hy^ — ex"" is solvable when n = 2a. For, assuming y = x^v, we find on substitution dv x :«« J-+6:C'V = CX% whence, dividing by a?^", we have x'-'f^+hv'=cx''-\ ax Now if n = 2a the above becomes whence dv dx an equation in which the variables are separated. If we restore to v its value ^ and transpose, this becomes x^dy — ayx'^'^dx ^ a-n ^ ,,x - — 'I ^ ^ ,, + cc" Wa;=0 (4), by: — ex V^v ART. 4.] THE FIRST ORDER AND DEGREE. 93 an exact differential equation, of wliicli the solution will be Ce-^^ - 1 or y=[ — Y]xtanW—- — b according as h and c have like or have unlike signs. 4. Secondly. The solution of the equation X -J- — ay ^- hxf = ca:;", is always reducible by transformation to the preceding case whenever — ~ — ^ = ^*. a positive inteo^er. For let y=^A-\ — , y^ being a new variable which is to replace ?/, and A a constant whose value is yet to be deter- mined. On substitution and arrangement of the terms we have -aA + hA'+(n-a + 2hA)- + h—,-^^=cx\..(5). Now let -aA^ lA^ = 0, then ^ = - or 0. These values of A we shall employ in succession. 5. First. If we assume A = j the above equation becomes 3/i yr 2/x dx Multiplying this equation by "~ and transposing, we have X 94 OF SOME REMARKABLE EQUATIONS OF [CH. TT. 'Now tills equation is of tlie same fo7'm as the given equation between y and x. The coefficients however differ, in that h and c have changed their places, and a has become a-{- n. And this transformation has been effected by the assumption a x" Hence, if in the transformed equation (6) we make a second assumption a + n x'' we shall have as the result ^%-(fl + ^")!/.+ i2/: = c^ (^)' h and c again changing places, and a + 7i becoming a + 2??. And the result of i successive transformations of the same series will be to reduce the given equation either to the form x-^-(a + in)^ji+C7/.'=hx'' (8), or to the form ^^-{^ + ^n)y, + hjt = cx'' (9), according as the integer i is odd or even. Xow by what has been established in Art. 3 the above equations will be integrable if we have n = 2 (a + in), an equation which gives 71 — 2a . .. -2^ = * (''^- 6. Secondly. If we assign to A its second value 0, (5) becomes ii-a) ~-\-h— --f^ = cx' ART. 7.] THE FIRST ORDER AND DEGREE. 95 Or, multiplying by ^ and transposing, irJ'-(«-a)2/. + cy,^ = te" (11). Now this equation for ij^ differs from tlie equation (G) obtained for y^ in the previous series of transformations only in that a in the coefficient of the second term has become — a. With this change only then that series of transformations may be adopted in the present instance. The change of a into — a in the final condition (10) gives 71 + 2a . — ^ = * as a new condition under which the equation in y is solvable. If 2 = 1 this gives n = 1a^ the condition first arrived at, and upon which the subsequent researches were based. Collecting these results together we see tliat the equation X -j— ay + hy"^ = cx'^ {s integrahle whenever — = — is aj^ositive ax JiiT, integer, 7. Let us now examine the form in which tlie solution is presented. If — - — -=L which is the condition arrived at in Art. 5, 2?i we have the series of transformations a x"" a + n and finally X Vi-x = h^— ^ + - » c ' y.' a A- In yz b «+ 0 \~\)n 96 OF SOME EEMAEKABLE EQUATIONS OF [CH. VI. where h='b or c, according as i is odd or even ; and tlie effect of these transformations is to reduce the given equation to one or the other of the forms (8) and (9). If in the above expression for y we substitute for y^ its value in terms oi y^, in that result again, for y^ its value in terms of y^ , and so on, we find a a + n a + 2n (A), the last denominator being ^-^ — —- -\ . The value of yi must then be determined by the solution of (8) or of (9), these equations being now susceptible of expression as exact ^liiferential equations in the forms "°"'''^^--^"+r)ff""''^" + a.-'-Va. = 0 (B), Cvj ~~ ox ^n.^y^ - („ + .•„) yX-'-Wo^ ^ ^„..-.^^ ^ , (C). Mlien therefore — = i a positive integer, the solution oftlie equation x — — ay ■\- hy^ = cx"^ icill he expressed in the form of a continued fraction hy (A), the value of yi in the last denomi- nator being given hy the solution of the exact differential equation (B) or (C) according as i is odd or even. Secondly, if — —— = i, which is the condition arrived at in An Art. 6, we have the series of transformations n — a a^ Vi = — + — • ART. 7.] THE FIRST ORDER AND DEGREE. 97 2n — a x"" y.. = ^^^4^ + | (12), where h = 'b or c, according as i is odd or even. From these, eliminating, as before, the intermediate variables y^, U-z^ "-1/^-1^ we find __^ ^ ~ n — a ic" c 2?i — a cc" b 3?i — a .^. ~~v~ ^^' ( 2. — 1 ) 7i — (Jj QC^ the last denominator being 1 — . In this case, however, the equation for y^ formed bj changing a into — a in B and G will be CVj —" Olio or '^' }; , ^^;p + ic"'^Wic = 0 (F) byi —ex according as ^ is odd or even. When therefore — ~— = i cc j^osttive integer, the solution of ^n X -~—ay + hy'^ = ex" is expressed by (D), the value of yt in the last denominator being given by the exact differential eq^uation (E) or (F) according as i is odd or even. Ex. Given cc ^^ — ?/ + ?/^ = a;^ B. D. E. 7 98 OF SOME EEMAKKABLE EQUATIONS. [CH. VI. 71 + 2a ^ ^., n-2a Here 7z = J, « = 1, and as —^ =2 wlnle -^^ = -1. the formula (D) and (F) must be employed. Assuming therein a = 1, Z> = 1, c = 1, n = f, ^ = 2, we have ,, = _fL, = _^^_ (13), 2/2 "being given by the exact differential equation iih:ZJmli^ + arid:c^O (14), from which we find 2// -J ilogf^) + 3^4 = C (15). The elimination of y, between (13) and (15) gives log"-'^"!-'":-^ + 6.i = C (16), which is the complete primitive. Ex.2. Given ^ + w' = aj'i ' ax This is an example of Kiccati's equation. Assuming there- fore u:='^,we find x~^--y + y'' = x^, which is identical with X ax ^ the equation last considered. Substituting therefore in (16) ux for 2/, we find after. reduction iog^"^!-'-"^:+6x*=c ^^^^_ 3Ma;= + 3 + iix' AET. 8.] GENERAL OBSERVATIONS. 99 General Observations, 8. The connexion between the two conditions for the dy solution of the equation x -j^ — ay -^ hy"^ = cjj", implied by the ,,,.., . n ±20, , . . _ double sign in the equation -— — = ^, may otherwise be established as follows. If the differential equation be written in the form 4l+^K^-f)=^"" ^''^' it becomes evident that it is symmetrical with respect to y and y — j . Assume then y — yas a new variable in place of y, and writing ?/ — j = y', y = y' + ji the equation becomes or x^-j^--\-aiy + hy"' = cx" (20), an equation which differs from the given equation only in that y has become y', and a has changed its sign. Hence the conditions 7i = -r-^ — and n = -—. are mutually dependent, 2l — 1 2i — 1 ^ r J and the value of y having been obtained for the former case, its value in the latter will be found by changing therein a into — ttj and finally adding j . It is here also to be noted that instead of beginning with x^ an assumption of the form y = A ■] as in Art. 4, we might x^ have commenced our reductions by the assumption y = ~ , the former of the above being proper for increasing by ?i, tiie 7—2 100 GENERAL OBSERVATIONS. [CH. VI. latter for dhuinislilng by n the quantity a. And as the first led directhj to the solution (A), so would the second have led directly to the solution (D). Lastly, it may be remarked that each of the above assump- tions is only the inverse of the other. To increase the value of a by 7i we had to employ the assumption ^i + — , which gives of and this indicates the form of the assumption for the case in which a is to be diminished. Hence also by admitting nega- tive as well as positive values of ^, the two forms of solution might be replaced by a single one. 9. We have seen in Art. 1 that Eiccati's equation ax is reduced by the assumption w = - to the form Hence the condition for the solution of Riccati's equation, found by substituting in the final theorem of Art. 6, 1 for a and m + 2 for «, will be 111 + 2 ±2 _ . 2m + 4 ~^' whence - = -2±^l (21), { being a positive integer. ART. 10.] GENERAL OBSERVATIONS. 101 We maj give to the expression for rii anotlier form, viz. — ^{ , , , m = —. , ^ admittinsr of the value 0 to2:ether with positive 2i± 1 ' ^ ^ ^ integral values. In order to prove this, let it be observed that two values of w included in (21) are -ii , -4(/-l)' m = —. — r , and m — 2^- 1 ' 2i- 1 If in the second of these values we change i — 1 into t, and therefore t into ^+ 1, a change which merely involves that we interpret i as admitting of the value 0 as well as of posi- tive inteofral A^alues, we find 'O' '"=2?n ('-)• "When 1 = 0 this gives m = 0, and as this value also results from the first of the expressions for m on making ^ = 0, we are permitted in that formula also to regard ^ as admitting of tlie same range of values. Hence, combining the two formulae in a single expression, we have "^ = 2711 (-')• i being 0, or a positive integer. 10. Riccati's equation may also be reduced, and it usually has been reduced, by a series of douUe transformations, of which the following will serve as an example. ^uation being We have The equation beini? -^ + l>u'^ = ca?"' let m = y- + —~ ^ ^ dx hx X u. du__^ 2 1_ du^ dx hx'' x'\t^ xhi'^ dx ' hic = ^— 2 + -^ + -T— 2 102 GENERAL OBSERVATIONS. [CH. YI. Substituting these values in tlie given equation, we have Whence, 1 In this equation assume x = z'''^^, then dx dz dx ^ ' dz ^ h x\' 1 xX du^ _ dx ■ cx' ^du. ""dx + CX "^-^V- -I- whence, after substitution and reduction ^+_^„^=_^rS^3 (24), dz 7?i -h 3 ^ m + 5 an equation differing from the given equation, as to its coeffi- cients and indices, in that h has been converted into , c m + 3' 1) . 711 -\- 4cL into , and m into ; but which is still of Riccati's form. The transformation, it will be observed, is a double one, as it affects the independent as well as the dependent variable. — j^i Now if m be of the form — — - , we find on substitution ^t ~~ J. and reduction m + l -4(^-1) m+3 2(i-l)-l* Hence, a second double transformation of the same nature as the last will reduce the differential equation to a form in which the index in the second member will become — —rr — ^-^ — - . And 2(z — 2)— 1 thus after a series of ^ transformations the index is reduced to 0, and the equation becomes solvable by separation of the variables. ART. 11.] GENERAL OBSERVATIONS. 103 To establish, another condition of solution, assume in the 1 -^ given equation w = - , aj = z'"^^ , then, after substitution and reduction, we have chj c + zrT-Ty = dz wi + 1 -^ m + 1 ' which, by what has preceded, will be solvable if we have m _ 4:1 4^ whence, m = — 2i+l Combining these results it appears that Eiccati's equation is — 4:1.. integrable if m = . , i being 0 or a positive integer. This agrees with (23). It is manifest from the complexity both of the transforma- tions above described and of the results to wliich they lead, that Eiccati's equation is, in its actual form, far less adapted for such transformations than the equation to which it is so easily reduced. 11. Eiccati's equation becomes linear on assuming _ 1 dw hw dx ' The transformed equation is ^t-5ca,»'«, = 0 (25). We shall consider it mider this form in a subsequent Chapter. 104 . EULER'S equation. [CH. VI. ToRiccati's equation some otliers of greater generality may be reduced "by a change of variables, e. g. the equation ^^Ix'^u^^cx'' (26), by assuming cc" Euhrs Equation. It has already appeared that the solution of a differen- tial equp.tion may sometimes be freed from transcendents introduced by integration. An example of this has been afforded in the instance of the equation dx chi + „, ■\.., =0, (Chap. II.), the solution of which is capable of being exhibited in an algebraic form, although immediate integration intro- duces the transcendental functions sin~^a.^, sin'^?/. The inquiry is here suggested whether in any otlier cases the direct inte- gration may be evaded, an inquiry the more important as our means of integration are so limited. Euler succeeded in ob- taining without direct integration the solution of the equation d^ , ^ n ^/[a-\-hx-{^ ex' + ex'' +/x^) \/{a + h7/ + cif + e/ +/]/*) and of some related forms. The result belongs to the theory of the elliptic functions, and may be established independently by the methods which more peculiarly pertain to that theory. But the method by which Euler arrived at that result demands notice here. 12. To integrate the equation ^^^ . ^ = 0 fn ^|{a-\■hx^cx'■\-ex^■^fx'y ^{a^ly-^cifA-eif^fif) ^ ^' Representing the polynomials a + hx + cx^-{- ex^ +fx*y and ART. 12.] euler's equation. 105 a + hy-\- cif + ey^ -\-fy^ by X and Y respectively, tvc have to integrate -i^+^ = 0 (2). The ordinary solution of this equation in the sense of Art. 5, Chap. I. would be [ dx r dy ^ but it is our present object to obtain an algebraical relation between x and y without performing the integrations above implied. Let -77-v^ = t^ then [_dx__ iv(X)-'' J = V(X), | = -v(r) (3). Also let a? + ?/=;;, x—y = q. AYe shall endeavour to form a differential equation in which p and q are dependent variables, and t the independent variable. From (3) we have J=V{X)-V{F) (4), | = V(A') + V(r) (5); therefore f^ = A'-r dt dt = 5^ + <-i>2 + i ci (Sjf + {^)-ix)=A^{'{y)+.p'(:x)} iy-x) + A,{-iy)+'j>"'(^)](2/-^y + A,{4,^{y) + '^{x)}(^-xy + &c....(12), wherein A , A , A , &c. are the coefficients of the successive powers of x, in the development of the function — — - in a series of the form A^x + A^x^ + A^x^ + &c. 108 euler's equation. [ch. yi. For let y = x-\- li, then, employing a well-known symbolical form of Taylor's theorem, d ^^ {e'^^+l)cl>{x) h- ~ dx 1 dx (fiix + ll)-^'" {y) + {y)=a-\-hj + cf-\-e7f-\-fy' (18). Eepresenting either member of (16) by dt and assuming t as an independent variable, substitute the values hence deter- mined for (/) (a;), ^' (ic), 0'" (ic), &c. in the theorem (15). There will result J = V(.^(a^)i. J=V(j x — y = qy by which it becomes 2 whence multiplying by -§ ^ and integrating therefore L ^~y J the integral sought. EXEKCISES. 2. x^-ay^f = x" . ax ax CH. VI.] EXERCISES. Ill 5. -^^ — %^ = 1x , ax 6. Assuming the copclitlons for the solution of Riccati's equation, Art. 9, investigate those under which the equation y- + ^icV = cic" is integrable. 7. Assuming the conditions, Art. 6, under which X -T- — ay + l>y^ = ex"" is integrable in finite terms, investigate those under which the equation is inteo-rable in finite terms. 8. Transforming the equation x -j- — ay -\- hif = cx^'^, by assuming ic"= ^, an integrating factor may be found by Art. 6, Chap. Y. 9. The equation -7- + lu^ = cx^' + -2 j inore general than G/X X Riccati's, is reducible to the form x -j^ — ay + h'y^— c V, con- sidered in Art. 3, by an assumption of the form u — ' . 10. Hence investigate the conditions under which the former equation may be solved. 11. The same equation may be reduced to Riccati's form by an assumption of the form y ^ Ax'^ -Y z^ {x) ^ followed by a transformation affecting only x. 112 EXERCISES. [CH. VI. 12. Integrate tlie equation dx cJi V (a + J^ + cx'^-V ex"- -\-fx') ^{a-\- by + cif+ e/ +/y "by the application of the theorem of Art. 13. Ie3. Deduce from that theorem the following expression for the value of a definite integral, viz. : 240 ( 113 ) CHAPTER VII. ox DIFFERENTIAL EQUATIONS OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE. 1. Eeferrinq to the general type of differential equations of the first order, viz. : fUvM-o, dx_ we have now to consider those cases in which -^ is so in- dx volved that the given equation cannot be reduced to the form ax already considered. Freed from radicals the supposed equation will, however, present itself in the form n-2 +...+ P« = 0 (1), where P^, P.^^...Pn are functions of a? and y. An obvious preparation for the solution of such an equation, is to resolve its first member, considered as algebraic with clii respect to the differential coefficient -,— , into its component factors of the first degree. If ^:>i, p^"'2'>n he the roots of (1) thus considered, we shall have ^_«.Vly-^l..ff^,0 = 0 (2), (1-^.) \dx V \dx B.D.E. 8 1 14 DIFFERENTIAL EQUATIONS OF THE FIRST [CH. VII. 2\ , i^2' • • -Pn l^eing supposed to be determined as known func- tions of X and y. And it is now manifest that anj relation between x and y which makes either one or more than one of the factors of the first member to vanish, will be a solution of the equation, and that no relation between x and y not pos- sessins: this character will be such. Hence if we solve the separate equations ^// ,, _o '^-79 -0 '^'^ -71 -0 f3^ d^-^'^-^^ dx ^'-'-^'"'dx ^--^ ^^^' any one of the solutions obtained will be a solution of (2), since it will make one of its factors to vanish. And if we express the different solutions thus obtained, each with its arbitrary constant annexed, in the forms r,-c,= o, t;-(7, = o,... F„-a. = o, any product of two or more of these equations will also be a solution of (2), since it will cause two or more of its factors to vanish. Ex. Given the differential equation Here the component equations are dif Tx^''^ = '^ and their respective solutions are \o^y-ax-c^=^0 (5)^ io^y -{■ax-c^ = 0 ,.... (6). Either of these equations is a solution of the given equation, and so is their product (logy-a^-cj (logy + aaj-g =0 (7). ART. 2.] ORDER, BUT NOT OF THE -FIRST DEGREE. 115 2. And here two important question.*? are suggested. First, how is it that two arhitrarj constants present themselves in the solution of an equation of the first order ? Secondly, is it possible to express with equal generality the solution of the equation by a primitive containhig a single arbitrary constant in accordance with what lias been said of the genesis of differential equations of the first order, Chap. i. Art. 6 ? These are connected questions, and they will be answered together. The equation (7) implies that y admits of two values each involving an arbitrary constant, but it does not imply that y admits of a value involving two arbitrary constants. The component factors of the solution separately equated to 0, as in (5) and (6), give respectively y^Cr, y^Cr- (8), each of which involves one arbitrary constant only, and each of which corresponds to a single factor of the given difterential equation. The true canon is, not that a general solution of an equation of the first order can involve only one arbitrary constant in its expression, but that each value of y which such a solution establishes involves in its expression only a single arbitrary constant. At the same time there is a real sense in which it remains true that every difterential equation of the first order, wliat- ever its degree may be, implies the existence of a complete primitive involving a single arbitrary constant, and there is a real sense in which such primitive constitutes the general solution of the difterential equation. To reconcile these seem- ing contradictions I shall shew that if we sup):)Ose the arbi- trary constants c^ and c^ in (7) identical, and accordingly replace each of them by c, we shall have an equation which will be, first the true primitive of (4), in that it will generate that equation by differentiation and the elimination of c, secondly its general solution, in that no particular relation is deducible from the solution (7) involving two arbitrary con- stants which may not also, by the use of a lawful freedom of interpretation, be derived from it. S— 2 116 DIFFEliEXTIAL EQUATIONS OF THE FIRST [CH. VII. Tims replacing c^ and c^ by c, we have (log y — ax — c) (log y-\-ax — c) = () (9) , whence (log yY— c^j? — 2c log y + c' — O. Differentiating, and representing -^ bj ^^^ 2 loo- ?/ ^- - 2cfx - 2c^ = 0, whence c = + io^: ?/. Substituting this value in (9), we have (^-?-)(?'--^)-' which reduces to aV(ay-/)=0. Or, rejecting the factor aV which does not contain ^, and replacing i:> bv -y- , ST-'V-- the differential equation given. Thus (9) is its complete primitive. Again, that solution is general. The two relations between y and x which it furnishes are y=Ce", y=Ce-" (10), and these differ in expression from (8) only in that the arbi- trary constant is here supposed to be the same in one as in the other, but as it is arbitrary and admits of any value, there is no single relation implied in (8) which is not also implied in (10). And it is in this sense that the generality of the solution is aflirmed. ART. 3.] ORDER, BUT NOT OF THE FIRST DEGREE. 117 3. These illustrations will prepare the way for the de- monstration of the general theorem which they exemplify. TuEORE^r. If the differential equation of the first order and d^ degree he resolved irito its comjyonent eciuations dii r. du ^ d'l and if the complete primitives of these equations are V^ = c^, V^= c,^,... J\ = c„, then the complete primitive of the given equa- tion icill be (T;-c)(T;-c)...(r„-c)=o. Let lis first examine the case in which the proposed diffe- rential equation is of the second degree, and therefore express- ible in the form (j--pM-7^-po) = 0. Suppose that the ^dx ^V \du; integral T"i = q is derived from the equation ~^—p^ = 0 by means of an integrating factor fx^ . Then d V^ = filj^ — 2\] dx. In like manner we shall have d^\ = juj-: — pA dx. Now taking the equation (T';-c)(r,-o)=0 (11) as a primitive, we have, on differentiating with respect to x and 1/, (T- -c)o (12). Therefore c = -—— rj^-^ , d]\-]-dV 118 DIFFERENTIAL EQUATION'S OF THE FIRST [CH. A^I. Substituting these values in (11), we have {v^-r,ydv,dr,^o (13), which gives (F,-F,)X;^,g|-2..)g-^,) = 0 (U). • And this, on rejecting the factor ( J\ — V^Y/Jb^fM^ which does not contain any differential coefficients, becomes identical with the given differential equation. Hence ( F^ — c) {V^— c) =0 is the complete primitive of that equation. To generalize this particular demonstration it would be necessary to eliminate c between the equation {r,-c){f\-c)...{V„-c)=0 (15), and the equation thence derived by differentiation with re- spect to X and y. The ordinary process of elimination, as exemplified above in the particular case in which 7i = 2, would be complex, but the result may be determined without dif- ficulty by logical considerations. It will suffice for this pur- pose to consider the case in which ?i = 3. We have then as the supposed primitive (F^_c)(F,-e)(F3-c)=0 (16), and as the derived equation (F.-c)(F3-c)f + (F3-c)(F.-c)5 + (F.-c)(F,-c)g = 0 (17). Now (16) implies that some one at least of the equations is satisfied. ART. 3.] ORDER, BUT NOT OF THE FIRST DEGREE. 119 If tlie first of these equations is satisfied we have c = V^, and substituting this value in (17) there results {V,-Vd{y,-V,)dV^ = 0 (18). If the second equation of the system is satisfied, we have in like manner (r;-iQ(i';-K)cn;=o (lo). If the third equation of the system is satisfied we have 0\-r,){r,-r,)dv, = o (20). Hence the existence of (16) as primitive supposes the exist- ence of some one at least of the equations (18), (19), (20), and therefore of the equation (v.-v.nv,-r.nv,-Krdv^dr,dv, = o (21), which is formed by multiplying those equations together. Conversely the supposition that the equation (21) is true, involves the supposition that one at least of the equations (18), (19), (20) is true. The equation (21) is thevehre equivalejit to the result which ordinary elimination applied to (16) and (17) would give. The same process of reasoning applied to the more general equation (15) as supposed primitive, would lead to a result of the form KdV^dK^...dV, = 0 (22), K being the product of the squares of the differences of V V V On comparison with (13) we see that in the particular case of n = 2, this is not only equivalent to but identical with the result of ordinary elimination in that case. And this identity of form, though it is not necessary to our present pm-pose to establish it, might be demonstrated generally. Now d F, = fi^ U^ - 2)}j dx, d i; = fi, H^ - i)}j dx, &c. 120 DIFFERENTIAL EQUATIONS OF THE FIRST [CH. VII. Hence (22) gives or, rejecting the factor KfM^/jL^.../jL^, which does not contain differential coefficients, Of this equation it has therefore been shewn, as was required, that ( J\ —c) {V^ — c) , . . {Vn — c) = 0 constitutes the complete primitive. Here the component equations are dy /«\4_ di/ f(f'\^_r. dx \x) ' dx \x) ' and their respective integrals are y-c^-2sl[ax) = 0 (2), y-c, + 2VM = 0 (3). Replacing both constants bj c and multiplying the equations together, we have [y-cy-4.ax = 0 (4), as the complete primitive. Now this primitive represents a series of parabolas, the parameters of which are constant and equal to 4a, and the axes of which are parallel to the axis of x] but the ver- tices of which are situated at different points of the axis of y, corresponding to the different values which may be given to the arbitrary constant c. Of these parabolas the equations (2) and (3), which may be written in the more usual forms y - c, = 2 ^Jiax), y - c^= - "1 ^{ax), ART. 4.] ORDEE, BUT NOT OF THE FIRST DEGREE. 121 represent respectively the positive and the negative branches, while the equation {y-c,-2^|[ax)][y-c,■\■2^/{ax)] = 0 (5), represents the terms which would be found bj taking one positive and one negative branch, hut not necessarily from the same parabola. Thus there is no portion of the loci re- presented by the apparently more general solution (5), which is not also represented by the complete primitive (4). The defect of generality, if as such it is to be regarded, consists in this that while each branch of every curve in the series is represented, tliose branches which belong to the same curve are paired together. 4. There are certain cases in which differential equations of the first order can be solved without the resolution of the first member into its component factors. Of these the most important are the following. 1st. When the given equation contains only one of the variables x and y in addition to -— , being either of the form uX or of the form ^i^'f:r'' ^(^-S)=«- 2ndly. When, involving x and y only in the first degree, it is expressible in the form x^ {p) + yf {p) = x^)^ ^'^lere 2^ = — * .Srdly. When the equation is homogeneous with respect to x and y. These cases we shall consider separately. 122 DIFFERENTIAL EQUATIONS OF THE FIRST [CH. YII. Equations involving only one of the valuables x and y with ,- . ax 5. In this case if, representing -^ by 2^1 ^^^ regarding ^ as a new variable, we form a differentieil equation between p and the variable which does not enter into the original equation, and integrate the equation thus formed, the elimina- tion of 2> between the resulting integral and the original equation will give the complete primitive required. For it will express a relation between x, y, and the arbitrary con- stant introduced by integration. Thus if from the equation F{x^ p) = 0 we deduce x =f{p>), then, since dy=pdx^ we have therefore yr= \pf\p) d]) + c (1). After the integration here implied y will be expressed as a function of p^ and c, and between that result and the original equation p must be eliminated. In like manner, if from F(y,p) = 0 we deduce y=f[pj), the equation dy=pdx gives f {p) dp=pdx, whence dxJ^^dp, whence /^^ + 0 (2), x p between which (after the integration has been performed) and the original equation, p must be eliminated. But these methods, though always permissible, are only advantageous when it is more easy to solve the given equa- tion, with respect to the variable x or y which it involves, than with respect to p. ART. 5.] OKDER, BUT NOT OF THE FIRST DEGREE. 123 Ex. 1. Given x=^l-{-]f. Here chj = pdx =px ^^dj^ = Zp^dp ; therefore y= r -^ <^ (3). Now as tlie original equation gives p=(x— l)^, the com- plete primitive found by substitution of this value in (3) will be y^l^{x-l)^+C (4), and it would be directly obtained in this form by integrating the original equation reduced by algebraic solution to the form dii , ^>i This example illustrates the process but not its advantages. Ex.2. Given a;=l +p+p^ Here dy —pdx =j)dp + 3^A//:> ; therefore y--^ ■\- ~r +c (5), between which and the original equation p must be eliminated. We may do this so as to obtain the final equation between x and ?/ in a rational form ; but, if this object is not deemed im- portant, we may, by the solution of a quadratic, determine^ from (5) and substitute its value in the given equation. Ex. 3. Given y^p'-\- 2/. Here since pdx = dy we have dx — - dy = 2 dp + Gpdj) ; therefore x = 2p + Sp^ + c. 124 DIFFERENTIAL EQUATIONS OF THE FIRST [CH. YII. From this equation we find _ -1 + \/(3a;+ C) ^~ 3 ' (7 being an arbitrary constant introduced in the place of 1 — 3c; and y will be found bj substituting this value of j:> in the original equation. Equations in which x andy are involved only in the first degree, the typical form being x(f) {p) +sy^ \p) = % {p). 6. Any equation of the above class may be reduced to a linear diiferential equation between x and p, after the solution of which, p must be eliminated. The reduced equation is found by differentiating the given equation and then eliminating, if necessary, the variable y. It may liappen that such elimination is unnecessary, y disappear- ing through differentiation. Ex. Let us apply this method to the equation y = xp+f{p) (1), usually termed Clairaut's equation. ^ Differentiating, we have ^ whence [x+f{p)}£^ = o, Kow this is resolvable into the two equations, ^+/{p)=0 (2), i- (^)- The second of these, which alone contains differentials of the new variables x and p, is the true differential equation between x and^. ART. 6.] ORDER, BUT NOT OF THE FIRST DEGREE. 125 Integrating it we have jp — c^ and substituting this value of^ in (1), y = cx-\-f{c) (4), which is tlie complete primitive required. But what relation does the rejected equation (2) bear to the given differential equation (1), and what relation to its complete primitive just obtained? If we eliminate^; between (1) and (2) we obtain a new rela- tion between x and y not included in the complete primitive already found, i.e. not deducible from that primitive by assigning a particular value to its arbitrary constant, and yet satisfying the same differential equation, and, as we shall hereafter see, connected in a remarkable manner with the com- plete primitive. Such a relation between x and y is called a singular solution. We shall enter more fully into the theory of singular solutions in a distinct Chapter, but the following example will throw some light upon their nature, as well as il ustratc the process above described. Ex. Given y = xj) + m Here differentiatinG: we have From the equation -f =0, we have ^; = c, whence y = cx + -^ (o), the complete primitive. From the equation x — 7. = 0, we have P y©. 126 DIFFERENTIAL EQUATIONS OF THE FIRST [CH. YII. and this value substituted in the original equation gives, after freeing the result from radical signs, ?/^ = 4wic (6), the singular solution. Here the singular solution (6) is the equation of a parabola whose parameter is 4m, and the complete primitive (5) is the wall-known equation of that tangent to the same parabola which makes with the axis of x an angle whose trigonometri- cal tangent is c. Xow, for the infinitesimal element in which the curve and its tangent coincide, the values of x^ y, and -j- are the same in both. And thus it is that the algebraic equations of the curve and of its tangent satisfy the same differential equation of the first order. On the other liand, if (5) be regarded as the general equa- tion of a system of straight lines, each straight line in that system being determined by giving a special value to c in the equation, the envelop or boundary curve of tlie system will be determined by (6). Here the singular solution is presented as the equation of the envelop of the system of lines defined by the complete primitive. 7. In the second place let us consider the more general' equation Differentiating, we have whence or dx _ f^{p)_ ^ ^ cf^'ip) ART. 7.] OEDER, BUT NOT OF THE FIRST DEGREE. 127 a linear equation of tlie first order by vv^liich x may "be deter- mined as a function of ij. The elimination of ^> between tlie resulting equation and the given one will give the complete primitive. The typical equation may be reduced to the above form by dividing by '^ {p), but it may also be treated independently by direct differentiation. Instead however of forming a differential equation between X and p, we may form a diffei-ential equation between y and p. Or, with greater generality, representing any proposed function of p by t, we may form a differential equation be- tween either of the primitive variables and t. Such a diffe- rential equation will necessarily be linear with respect to the primitive variable retained, and its solution must of course be followed by the elimination of t. And this general procedure, more fully to be exemplified when we come to treat of some of the inverse problems of Geometry and of Optics, is often attended with signal advantage. Ex. Given x + yp = ap^. We shall reduce this to a differential equation between x and ^9. Differentiating, we have then eliminating y by means of the given equation, we have wliich may be reduced to the linear form dx _ op dp p{\+f) 1+/' its integral being 128 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. [CH. YII. If in tliis equation we substitute ior j^ its value in terms of x and y furnished by tiie given equation, i.e. if we make ^'~ 2a we shall be in possession of tlie complete primitive. Had we chosen to form a differential equation between y and ^, we should have, on differentiating the given equation while regarding y as the independent variable, dx 1 whence, replacing ^ by - and reducing, J J. dy p _ lai?' dp H-;/^~ \-^f ' therefore on intesrration from which, as before, p must be eliminated. The final results are of course identical. Homogeneous Egiiations of the first order. 8. Equations which are homogeneous with respect to x and y may be prepared for solution by assuming y = vx. The typical form of such equations is «=>(|.i')=0 (!)• Assuming tlien -= r, and dividing hj x", we have ^{■o,p) = 0 (2). ART. 8.] HOMOGENEOUS EQUATIONS. 129 If we can solve this equation with respect to ^, we have But, since y = xv dv Thus the transformed equation becomes dv ^ /-/ N , dv dx whence 2^-^ ^ = ^> v-f{v) X an equation in which the variables are separated, and in the integral of which it will only remain to substitute for v its value - . X But if it be more easy to solve (2) with respect to v than with respect to p, and if the result be then restorino; to v its value - , we have ^ X which is a particular case of tlie equation of the previous section. Hence differentiating, we have P=f{p)+¥'{p)% from which results B. D. E. ^ fip) -P ' 130 HOMOGENEOUS EQUATIONS. [CH. YII. an equation in which the variables x and ]) are separated. Between the integral of this equation and the given equation f must be eliminated, and the relation between x and y which results will be the complete primitive. Ex. Given yp + nx = JsJ{y' -|- nx"") ^/[l +/). Assuming y = vx, we have vp + n = V(v' + w) V(l + /)» the solution of which with respect to p gives But P^^dx'^'"' Therefore ^ J = ± a/^"^^ ^^(«' + «)' Integrating, we have dv I (n — 1\ dx }?-\-n)~ ~\/ \ n ) X ' log{^ + V(^^ + ^)} = ±y('^)log^ + ^; therefore v + V(^' + ?0 = cst'^^ , or, replacing t; by - , y + ^J{f■\■nx') = cx'^''^^^''^^' the complete primitive. ART. 9.] EQUATIONS SOLVABLE BY DIFFERENTIATION. 131 Equations solvable hy differentiation, 9. A remarkable class of equations, the theory of which has been fully discassed by Lagrange, deserves attention. It has been shewn. Chap. I. Art. 9, that if two differential equations of the first order, each involving a distinct arbi- trary constant, give rise to the same differential equation of the second order, they are derived from a common primitive involving both the arbitrary constants in question. Let us suppose these differential equations of the first orde; to be reduced to the forms '^(^'^'^sr" «' ^(-'^'2)=^ c^)' and let the primitive obtained by the elimination of -~ be ^ (x, y, a, h) = 0. Lagrange has then observed that if wv have any difterential equation of the first order of the form 'H*'2''£ )' ^(^'^'2)}=° ^'- its complete primitive will still be O {x, y, a, h) = 0, but with the condition that a and h are no longer independent con- stants, but are connected by the relation F{a,h) = 0. This is an obvious truth. For as, by hypothesis, the su])- poscd i)rimitive <3p {x, ?/, a, b) =0 gives ^ (^^ 1/, ^ = cc, ^Uy/£) = h it will convert (3) into F {a, b) = 0, and will therefore sati\/i/ that equation if a and b are connected by the relation F{a,b) = 0. 132 EQUATIONS SOLVABLE BY DIFFEEfENTIATIOX. [CH. YII. Moreover it contains virtually only one arbitrary constant, for the relation F{a, h) = 0 permits lis to determine h as a function of a. Hence it will constitute the complete primitive of (3). See also Chap. L Art. 10. This result may he expressed in the following theorem. If any differential eqiLation of the first order he expressible ill the form F{cl,,ir) = 0 (4), where (j) and ^fr are functions of x, y, -i- , such that the dif- ferential equations (\> = a, -^ = Z>, are derivable from a single 'primitive involving a and h as arlitrary constants, the solution of the given differential equa- tion loill he found hy limiting that primitive hy the condition F{a,h) = 0, so as actually or virtually to eliminate one of the arbitrary constants. Ex. Suppose that the given equation is Vl'+©}=4^*''i) <"• IS'ow the diflferential equations of the first order '» + !'£— (-'• Vl'-©]=' ■■ «• are derivable from a common primitive; for, on difi"eren- tiating them, we have respectively , fdyY d'^y ^ dy dx (^ /di/V dSA ^ ART. 9.] EQUATIONS SOLVABLE BY DIFFERENTIATIOX. 183 and these agree as differential equations of the .second order, Chap. I. Art. 9. That common primitive, found Ly elimi- dv nating -j- between (2) and (3), is Hence the primitive of the given equation is y + (a. -ar ={/(«)- (4). We might also proceed as in the solution of Clairaut's equation. Differentiating the given equation, we have VI-©] ^--^'l"+2' + ylr = 0 (2), where <^ = — , '^—y'^ — V]^^- Now ^'p — a^ y^ — ypx = h, are derived from a common primitive y^ — ax^ = 5. The solu- tion of (2) will therefore be, y^ — adi? = 1) with the connecting relation between the constants^ Aal + Ba + h = 0. And this will be found to agree with the previous result. EXEECISES. The following examples are chiefly in illustration of Arts. 1, 2, 3, 5. 1. \dx/ \dx) 2. 3. fdy^ \-x \dx) ~ X CH. VII.] EXERCISES. 4. 5. '-'t*^m- 6. "•hKif- 7. *=-iVl-(S]- 8. "»SV{'-©]- 9. i-V{-©]=»- 10. .{■.(i)-f-..=.. 11. \dxj x' + 2aaj * 137 The following examples are intended to illustrate Art. 6. The singular solutions as well as the complete primitives arc to be determined. 12. y = x^ + ^-(^]\ •' dx dx \dxj ' '"■ '-SVl'--'©]- The following examples are in illustration of Arts. 7 and 8. 14. .-4lWh(i. 138 EXERCISES. [CH. VII. 15. y^x-J + ax '«• -'2='©' The following examples are in illustration of Art. 9. 18. .i=-/|/-w| r ^ •in 1 ^ . ^^^ ^ ». ,-..|=/Hi)}. ( 139 ) CHAPTER YIII. ON THE SINGULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. *° 1. In the largest sense whicli lias been given to the term, a singular solution of a differential equation is a relation between the variables which reduces the two members of the equation to an identity, but which is not included in the com- plete primitive. In this sense, the relation obtained by equating to 0 some common algebraic factor of the terms of the equation might claim to be called a singular solution. But, in a juster and more restricted sense, a singular solution of a differential equation is a relation between x and y, which satisfies the differential equation hy means of the values wJiich if gives to the differential coefficients ^ , -j-^, &c., but is not included in the complete primitive. In this sense the equa- tion x^ + y^ = n^, is a singular solution of the differential equa- tion of the first order It reduces the members of that equation to an identity, but not by causing any algebraic factor of them both to vanish. xVt the same time it is not included in the complete primitive y — cx = n\f{\-\-(r). And this is the juster definition, because that which is essential in the sin.2:ular solution is thus in a direct manner 140 ox THE SINGULAR SOLUTIONS OF [CH. YIII. connected with that whicli is essential in the differential equation. Def. Chap. I. When it is said that a singular solution of a differential equation is not included in the complete primitive, it is meant that it is not deducible from that primitive hj g'iving to the arbitrary constant c a particular constant value. But although a singular solution is not included in the complete primitive, it is still implied by it. Upon the possibility of satisfying a differential equation by an infinite number of particular equa- tions, each formed by the particular determination of an arbitrary constant, rests the possibility of satisfying it by another equation, to the formation of which each particular solution has contributed an element. We have seen in Chap. VII. how a singular solution, as representing the envelope of the loci defined by the series of particular solu- tions, possesses a differential element common with each of them. We shall now see that this property is not accidental — that it is intimately connected with the definition of a singular solution. It is important that the two marks, positive and negative, by the union of which a singular solution of a differential equation of the first order is characterized, and by the expres- sion of which its definition is formed, should be clearly appre- hended. 1st. It must ffive the same value of -^ in terms of x ^ dx and y, as the differential equation itself does. This is its positive mark, a mark which it possesses in common with the complete primitive, and with each included particular primi- tive. 2ndly. It must not be included in the complete primitive. This is its negative mark. Upon the analytical expression of these characters the entii'e theory of this class of solutions depends. Among the different objects to which that theory has reference, the two following are the most important. 1st. The derivation of the singular solution from the complete primitive. 2ndly. The deduction of the singular solution from the differ- ential equation without the previous knowledge of the com- plete primitive. The theory of the latter process is so de- pendent upon that of the former that it is necessary to consider them in the order above stated. ART. 2.] DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 141 Derivation of the singular solution from the comjctlete primitive. 2. The complete primitive of a differential equation of the first order, whatever may be the degree of the equation, is of the form j> {x, ?/, c) = 0. If we give to c a particular constant value in this equation we obtain a particular primitive. If we give to c a variable value by making it a function of cc, or of?/, or of both, we, as will immediately be shewn, convert the equation into any desired relation between x and y. We propose then to deter- mine c as variable, but as so varying that the resulting relation between x and y shall continue to satisfy the differ- ential equation. The general effect of the conversion of c into a function of X or of y must first be considered. Prop. I. A primitive equation ^{x, y, c)=0 may, hy the conversion of c into a function of x he transformed into any desired equation containing x and y together, or y alone, hut not into an equation involving x without y. Let the desired result of transformation be ^(^, 3/)=^^ orx(?/)=0, involving y at least. Combining either of these equations with the primitive we can eliminate ?/, and so obtain a rela- tion between x and c which will determine c as the function of x required. It is evident however tliat the conversion of c into a func- tion of X could not convert the primitive into an equation not involving y. For a variable cannot be eliminated from an equation, except by the aid of another equation which contains that variable. 142 DERIVATION OF THE SINGULAR SOLUTION [CH. VIIL Similarly the conversion of c into a function of y would enable us to convert the given primitive into any desired equation involving, of the two variables, at least x, Ex. Let it be required to convert the equation y = cx into x^-iry'^=lj by the conversion of c into a function of x. Eliminating y from the given and the proposed equation, we have a;^+cV=l ,2\ whence c = — — — -. x This value of c substituted my — ex, converts it into which is equivalent to a?^ + ^/^ = 1. 3. Let us now enquire what determination of c as a func- tion of X will convert the primitive dc ' d(f) dx ...(9), and the second, to 0. members of these equations must be equated We see that these second members will usually vanish if ~ = 0. And this equation -^ = 0 is adopted by some writers as a sufficient expression of the rule for the derivation of the singular solution from the complete primitive, unrestricted by any accompanying condition. (Lagrange, Calcul des Fonc- tions, p. 207). We must notice however that the vanishing oi -J- or -J- in (9) may be due not to the vanishing of the numerator -,^ , but to the assumption of an infinite value by the denominator -^ or -~ . The latter is indeed quite as dy ax ^ ])robable a cause as the former when

= 0, is rational and integral ice may for convenience employ the single condition ^ = 0; but never without reference to the fundamental con- dc ' J J ditions (1). In the statement of tlie above theorem the two following particulars should be noticed. 1st. It supposes c to be determined as a variable quantity. Now if c be obtained as a function of both x and t/, as it generally will be if the condition -j- = 0 be made use of, it may be necessary by a subsequent elimination to reduce it to a function of one of the variables, in order to assure ourselves that it is not constant in virtue of the relation between x and y established in the primitive. 2ndly. The theorem takes account equally of the positive and of the negative characters of a singular solution. The existence of a variable value of c determined by either of the conditions (1) does not assure us that the resulting solution is singular, unless constant values of c are at the same time excluded. Ex. 1. The equation ?/' - "Ixy 'J^ + (1 + ^') (f^T= 1» ^^^ for its complete primitive y = cx-{-^{l — c^). Its singular solution is required. Here -^ = x 77- ^ . Hence / =0 frivQa for c the dc V(l - c ) ^^c ^ variable value c = —rr^> — r^ , the substitution of which in the primitive gives y = V(-«' + i) (!)• 150 SINGULAR SOLUTIONS OF [CH. VIII. This value of ?/ satisfies the given differential equation, and it is evident on inspection that it is not included in the com- plete primitive. Formally to establish this, we find on elimi- nating y between that equation and (1) solving which, with respect to c, we have the unique value e = — — — — , which, agreeing with the value of c before \/ [X -]- 1) employed, shews that c admits of no other value, and in particular that it admits of no constant value. The solution is therefore singular. (Ix The condition -7- = 0 would, in the above example, give c = -^ , and lead to the same final result. We must be careful not to rely upon the condition -— = 0, except under the circumstances specified in the general theorem. This remark will be illustrated in the following example. Ex. 2. The complete primitive of the differential equa- tion y =i^x + — , where ^ stands lor — - , is y — ex = 0, and, if we represent its first member by ^, the elimination of d^ dc tion y"^ — Amx. c between the equations ^ = 0, -^ = 0, gives the singular solu- But, though this is not a procedure likely to be adopted, if we reduce the primitive by solution to the form ^ -^^^ L - 2c = 0, X ' ART. 5.] DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 151 and then represent its first member by , we shall have dy _ d^ d dc is seen to be dependent, not upon the vanishing of - , but upon the assumption of an infinite value by — . The true ground of preference for the conditions r = ^j dx . -T- = 0, consists, however, not in the directness of their appli- cation to irrational forms of the primitive, but in the plainness of their geometrical interpretation, and still more in their fun- damental relation to the problem of the derivation of the singular solution from the differential equation — questions hereafter to be discussed. The following example is intended to illustrate that portion of the theorem which relates to the negative character of a singular solution. Ex. 3. The complete primitive of the differential equation is y = c[x — cy. The singular solution is required. Here the condition -v- = 0 gives dc {x -c){x- 3c) = 0, lo2 SINGULAR SOLUTIONS. [CII. YIII. Avhence c = x^ ^^ o • These values of c, both of which a. variable, reduce the primitive to the forms 2/ = 0, 2/ = ^, and both these are solutions of the differential equation. Bu while the latter of the two is not included in the complet primitive, the former is included in it. If between the equa tions y = c{x-cy, y = 0, we eliminate ?/, the resulting values of c will be c = 0, c=x. We see therefore that the solution to which we were 1 bj the assumption c = x is a particular integral. But it pc sesses tlie geometrical properties of a singular solution e. plained in the following Article. Geometrical Interpretation. 6. Let y =/ [x, c) represent a fomily of curves the indi vidual members of which are determined by giving difFeren values to c. Then, adopting for a moment the language o infinitesimals, the differentiation of y with respect to c implies the transition from an ordinate y of one curve to an ordinate y + -,- dc, corresponding to the same value of x, but belonging to another curve of the series; viz. the curve obtained by changing c into c + dc. When we impose the condition ;j- = 0, we demand that this transition shall not affect the value of the ordinate y corre- dii sponding to a value of x determined by the equation -/ =0, AKT. 7.] GEOMETRICAL INTERPRETATION-. 153 Hence the singular equation obtained by the elimination of c between the e(^uations y=f{x,c), y=0, represents the locus of such points of successive intersection. In stricter language, the singular solution represents the locus of those points which constitute the limits of position of the points of actual intersection of the different members of the family of curves represented by the equation ?/=/ (^, c\ always excepting the case in which that locus coincides with a particular curve of the system. (ill And as at these limiting points the value of -j ^^ ^^^ '^^'^^ for the locus of the singular solution and the loci of primitives, it follows that the former has contact with every curve of the latter system which it meets. The locus of the singular solu- tion is seen to be the envelope of the loci of primitives. The envelope of the loci of primitives is the locus of a singular solution, except when it coincides with one of the particular loci, of which it forms the connecting bond. Similar observations may be made with reference to the condition -7- = 0. ac Derivation of the singular solution from the differential equation, 7. We have found that the singular solution of a differen- tial equation considered as derived from its complete primitive possesses the following characters. 1st. It satisfies one of the conditions ^- — 0, -, =0. dc dc 2nd. It is not possible to deduce it from the complete primitive by giving to c a constant value. It has also been shewn that the positive conditions are equivalent except when the singular solution involves only one of the variables in its expression. 154 DERIVATION OF THE SINGULAR SOLUTION [CH. VIIL Now we shall endeavour to translate the above characters from a language whose elements are x, y, and c to a language whose elements are x, y, and ---- , — froni the language of the complete primitive to the language of the differential equation. If we differentiate with respect to x the complete primitive expressed in the form 2/=/(^.c) (1), we obtain the derived equation ^='^ •••(^)' and substituting in this for c its expression in terms of x and y given by the primitive (1), we have finally the differential equation in the form i> = ^{^,y) (3). Thus the differential equation (3) is the same as the derived equation (2), provided that c be considered therein as a func- tion of ic and y determined by (1). Accordingly we have • |i„(3)=|in(.)x|in(l), or ^P ir^ (s) =.^l^^A ^^f^L^ . dy ^ dxdc ' dc * . ,,v dc ^ dy ^ df(x, c) smce m ( 1) — = 1 -^ -f = 1 -^ •' , . ' ' ay dc do or finally | = il°st W' ART. 7.] FROM THE DIFFERENTIAL EQUATION. 155 provided that the value of the first member be derived from the differential equation, that of the second member from the complete primitive. In like manner if we suppose the complete primitive ex- pressed in the form we shall have through symmetry the relation, d /IN d , dx ,^. TxKp) = Ty^°^Tc (^)' the first member referring to the differential equation, the second to the complete primitive. The equations (4) and (5), which are rigorous and funda- mental, establish a connexion between the differential equa- tion and the complete primitive, and it now only remains to (£fi doc introduce the conditions -^ = 0, -j- = 0. We beffin with the dc ' dc * former. dii We have seen that when -,— = 0 leads to a sin ovular solu- > do ^ tion it does so by enabling us to determine c as a function of a?, suppose c = X. Before proceeding to more general considera- tions it will be instructive to make a particular hypothesis as dy to ih^form of the equation -^ = 0. Suppose then this equation to be of the form Q{c-Xr = 0 (6), m being a positive constant and Q a function of x and c, which neither vanishes nor becomes infinite when c = X. This hypo- thesis is at least sufficiently general to include all the cases in which -/ = 0 is algebraic. 156 DEEIYATIOX OF THE SINGULAR SOLUTION [CH. VlII, By (6) we have then dQ dX dp d , dii dx dx /„, dy dx "^ dc Q c-X ^ ' and the second term of the right-hand member liavnig c — X for its denominator and not containing c at all in its nume- rator, is infinite. At the same time, we see that no such infinite term would present itself were c determined as a con- stant. For Iet|=<3(c-«r,the„^logJ = g.ft the right, hand member of (7) being now reduced to its first term. The conclusion to which this points is that -j- is infinite for a singular solution, but finite for a particular integral. Again, suppose the value of c in terms of x and y fur- nished bj algebraic solution of the complete primitive to be c — 4>(x,y), then substituting this value in the equation c — X= 6, we obtain the singular solution in the form cl>{x,y)-X=0. Now the same substitution gives to the infinite term in the value of -/ the form dy dX ^^ (8). {x,y)-X "\Ye see then, in the case of a singular solution correspond- ins: to a determination c = X, that -y- as derived from the dif- ferential equation becomes infinite owing to (.t, y) — X occurring in a denominator. And, whatever modification of form may be made by clearing of fractions or radicals, we may still infer that, if u = 0 be a singular solution derived from an ART. 7.] FROM THE DIFFERENTIAL EQUATION. 157 algebraic primitive, the function -^ will become infinite, owing to u presenting itself under a negative index. The analysis does not however warrant the conclusion that di) any relation between x and y which makes y- infinite will be a solution. If m be a negative constant, the second term in the expression of y- is still infinite, but the prior condition --'^ = 0 is no longer satisfied. All we can affirm is that if -f-z= cc ffives a solution at all it will be a sin<2rular solution, dy "" ^ dx \ . , . Since ■^— = - , it is evident that a sin^ijular solution ori^^Inat- dy p ° ° ing in a determination of c in the form c= Y will make -7^ ( - ) infinite. dx \pj A contrast between the conditions f^ = 0, -,'- = 0, and the do dc conditions y = co , -r- ( — j = co , is also developed. The former lead to solutions, but not necessarily to singular solutions ; the latter do not necessarily lead to solutions, but when they do, those solutions are singular. Ex. 1. G iven ^ - 2xp + 2y = 0. Here p=x ± \/{x^ — 2y) , which becomes infinite if ?/ = '— , and this satisfies the difior- cntial equation. It is therefore a singular solution. It may be objected against the above reasoning, not only that it involves an assumption as to the form of the equa- 158 DERIVATION OF THE SINGULAR SOLUTION [CH. VIH. tion -r = 0, but also that it takes no ^account of any pos- ac sibilities arising from the first term in the expression of -^. But it serves well to illustrate what, in the vast ma- dy jority of instances, is the actual mode of transition from the one set of conditions to the other. We proceed to consider the question in a more strict and general manner. 8. AVhen -/- = 0 determines c as a function of x, it recipro- cally determines ic as a function of c, so that if a definite value be given to c, a corresponding definite value or values will be given to x. Let -y- be represented by yjr [x, c) , then dp _ d . dy dy dx ^ dc = li„,it ^f]2gi±+A£J^i}2E±lM (9), h approaching to 0. Now for a singular solution ^fr {x, c) = 0, and this being, from w^hat precedes, satisfied only by definite values of a?, cor- responding to our assumed definite value of c, it follows that •y^r [x-\- 7i, c) will not be equal to 0 for any continuous series of values of h however small ; neither then will log i/r (a? + ^, c) retain continuously the value of \og'\^r{x, c), viz. — oo . Thus the numerator of the fraction in the second member being equal to the difi'erence between a finite and an infinite quantity is infinite, and the limit of the fraction therefore infinite. Hence we conclude that a singular solution considered as derived from the primitive by the conversion of c into a function of X, satisfies relatively to the difibrential equation the condition dy And in the same way it may be shewn that a singular solu- tion derivable from the primitive by the conversion of c into a function of y satisfies the condition -y- f - j = cc . ART. 8.] FROM THE DIFFERENTIAL EQUATION. 159 Changing the order of the enquiry, let us now examine whether there exist any other forms of solution satisfying the condition -^ = co, -=- l-]=cc . If there be, it will be made evident that more is involved in the definition of a singular solution than we have yet recognized in our processes of deduction, or else that the definition must be enlarged. Expressing the condition -^ = cc , in the form tM=- (-)' we observe that it can be satisfied only in one of two ways, viz. either independently of c, or by some determination of c, and if the latter again only in one of two ways, viz. either by the determination of c as a function of x, or by the determina- tion of c as a constant. We may pass over the case in which the above equation is satisfied independently of c, because the relation obtained would involve x only, whereas it has been shewn that -^ = GO leads only to solutions involving y at least. We may also pass over the case in which it is satisfied by the assumption c = X, because such a value of c, if it lead to a solution at all, can only do so by satisfying the condition — = 0, and thus lead to the form of singular solution already investigated. There remains only the case in which the equation (10) is satisfied by a constant value of c. Let then the equation (10) be satisfied hj c = a. The most general assumption we can make respecting the form of its first member is the following, viz. ^og-f=cl>{c)f{x,c), dx "=" dc where (/> (c) is a function of c which becomes infinite when 160 DERIVATION OF THE SINGULAR SOLUTION [CH. VIIL assumes the constant value in question, and -^ {x, c) does not become infinite for such value. Hence the most general ^^ dc form of log ~ is log ^ = J (^) t (-^^ c)dx=^ (c) jyjr {x, c) dx. To give to this expression the utmost generality, we must, on effecting the integration with respect to ic, add an arbitrary function of c. Thus we shall have log ^ = (c) |j t (^, c)dx-Vx W| • Therefore ^ = 6'^(c){/^(^, ^M^r+xCO}, ac or, representing the function /'v/r(a:, c) dx + %(c) by ^{x, c). This is the most general form of -V^ , as determined from the primitive, which is consistent with the hypothesis that ,- 102: 1 becomes infinite for a constant value of c. Ac- dx ^ dc cordingly if, supposing the primitive to be given, we sought to determine the singular solution by the condition -^ = 0, we should be led to an equation of the form or ^(c) ^(a^, c) = -cc (12). Xow this equation is not satisfied by any value of c which makes ^ (c) infinite, unless it give to ^ {x, c) an opposite sign ART. 9.] FROM THE DIFFERENTIAL EQUATION. 161 to that of (/) (c). But this indicates in general the existence of a relation between x and c. Thus suppose <^ (c) = c, ^ (cc, c) = X, Then (12) becomes cx = — ^ ^ which demands that c should receive the value -co or + go according as x is positive or negative. In either case c is constant, but it is a dependent constant — dependent for its sign upon the sign of x. Thus the condition -^ = co may indicate the existence of a species of singular solution derived from the complete primitive bj regarding c, not as a conti- nuous function of x, but as a discontinuous constant, the law of its discontinuity being however such as to connect it with the variations of x. Ex.2. Given ^> = 2'-My. Here we find | = i(i + i°s^) ^^^)' which is infinite if y = 0. And this proves on trial to be a solution of the diff'erential equation, the true value of the indeterminate function in the second member when ?/ = 0 being 0 (Todhunter's I)if. Cal. Art. 158). Now the complete primitive is ?/ = e'^. Hence we see that ?/ = 0 is not a particu- lar integral in the strict sense of that term. Tiie value to be assigned to c is not icholly independent of x. We may there- fore regard y=0 as a singular solution satisfying the condition f = 00. dy 9. We have said that, in general, the equation (12) in- dicates the existence of a relation between x and c. A case of exception however exists. Eepresenting (a?, c), expressed in terms of x and C, to be capable of development in descending powers of C: suppose, too, that B. D. E. 11 .162 'DERIVATION OF THE SINGULAR SOLUTION [CH. VIII. the first term of the development is of the form A C\ where A is constant and r> — 1. Then as C approaches infinity, (12) tends to assume the form indicating that G, and therefore c, possesses more than one value, real or imaginary. Here, then, the condition -^ = go would accompany a solution possessing this singularity, viz. that it corresponds to a multiple value of c, the arbitrary constant in the complete primitive. It is in fact a species of multiple particular integral. Ex. 3. Given jp^ —pxy + y^ log ^ = 0. T-r ^V + ?/ \/(^^ — 4 loO^ ?/) Here p — — —. ^-"^ ; therefore dp ^x±>^{x^-^\o^ji) 1 . . dy 2 "^V(^'^-4iog^) ^ ^' and this is made infinite by ?/ = 0 and by a?^ — 1 log ?/ = 0, that is by y = ^^ y= e ' • Both these satisfy the differential equation, and the second is obviously a singular solution. To determine the nature of the first let it be observed that the complete primitive is y = e^^\ and that this reduces to ?/ = 0, irrespectively of the value of x, by the assumptions c = + co and c = — go . Now this is the only case in which two particular integrals agree. We might in any case, by changing in the complete primitive of an equation c into c^, get two values of c for a particular integral, but then it would be for every particular integral. It is only when the property is singular, tliat the condition -^ = co is satisfied. ART. 10.] FROM THE DIFFERENTIAL EQUATION. 163 It is obvious that one negative feature marks all the cases in which a solution involving y satisfies the condition -/- = co . It is, that tlie solution, while expressed by a single equation, is not connected with the complete primitive bj a single and absolutely constant value of c. In the first, or as it might be termed envelope species of singular solutions, c re- ceives an infinite number of different values connected with the values of a; by a law. In the second it receives a finite number of values also connected with the values of ic by a law. In the third species it receives a finite number of values, determinate, but not connected with the values of x. If we observe that all the above cases, while agreeing in the point which has been noted, possess true singularity, we shall be led to the following definition. Definition. A singular solution of a ditTcrential equation of the first order is a solution, the connexion of which with the complete primitive does not consist in the giving to c of a single constant value absolutely independent of the value of X. Criterion of species. 10. It is a question of some interest to determine whether a given singular solution, w = 0, of a differential equation, is of the envelope species or not. On the particular hypotheses assumed in Art. 7, it is shewn that singular solutions of the envelope species possess the fol- lowing character, viz. \i u = 0 be such a solution, then -/- ay becomes infinite through containing a term in which u is presented under a negative index. Now inquiries which are scarcely of a sufficiently elemen- tary character to find a place in this work, indicate (with very high probability) that this character is universal and indepen- dent of any particular hypothesis, and that it constitutes a criterion for distinguishing solutions of the envelope species from others. 11—2 1G4 CRITEPJOX OF SPECIES. [CH. VII I. As an example of an hypothesis different from that of Art. 7, let us 'suppose iy Q dc log (c — A)' rhich vanishes when c = X, We find dQ dX d , dij dx dx rf^'°Srfc-~+~ -A-)log(c- 'X) The second term in the right-hand member becomes inde- terminate when c = X, but its true value is oo , and it assumes this value in consequence of c — X presenting itself with a nearative index. We remark that the fraction , -. ^77 is log(c-A) one which vanishes with c — X in icliatever manner c— X ap- ])roaches to 0, — a consideration which is quite of essential importance. Applying the above criterion to some of the previous ex- amples, we see from the form of ^ in Ex. 1, Art. 7, that the singular solution belongs to the envelope species ; in (13) Art. 8, it is implied that the solution is not of that species ; in (14) Art. 9 two species are indicated, the solution ?/ = 0 resulting from log?/ = — 00 being not of the envelope species, while the other solution is of that species. 11. The collected results of the above analysis are con- tained in the following theorem. Theorem. The singular solutions of a differential equation of the first order (Def. Art. 9) consist of all relations which helong to one or both of the following classes, viz, 1st. Relations involving y, with or without x, which mahe dj) -j- infinite and only infinite, and satisfy the differential equation. ART. 11.] EXAMPLES OF SINGULAR SOLUTIONS. 165 2nd. Relations mvolvinrj x, with or without y, ivhich mahe -T-\-] infinite and only infinite^ and satisfy the differential equation. When a solution as al>ove defined is actually ohtained hy equating to 0 a factor which appears under a negative index in the expression of -J- or -j—l-j it may he considered to lelong to the envelope species of singular solutions. In other cases it is deducihle from, the complete primitive hy regarding c as a con- stant of multiple value, — its pt articular values heing cither \st dependent in some loay on the value of x, or Indly independent of X, hut still such as to render the property a singular one. We may add that there exist cases in whicli the characters of different species of solutions seem to be blended together. dr) • . Thus ~ may admit of both a finite and an infinite value, indicating a duplex genesis of the solution from the complete primitive. It may also happen that the assumption of an infinite value by -~- may be attributed, indifferently, either to a negative index or to a logarithm. And then it sliould be inquired whether or not the solution is of the envelope species, but marked with some peculiarity arising from a breach of continuity in the mode of its derivation from the complete primitive. The following examples are intended to elucidate particular points either of theory or of method. Ex. 1. Given (1 + x') (j^X- 2xy 'j^ + y'-l = 0. This equation, first discussed in Brooke Taylor's Methodus Incrementorum, is remarkable as having afforded the earliest instance of the actual deduction of a singular solution from a differential equation (Lagrange, Calcul des Fonctions, p. 276). We shall first explain Taylor's procedure, and afterwards apply the above general Theorem. 1G6 EXAMPLES OF SINGULAR SOLUTIONS. [CII. VIII. Taylor differentiates the equation, and finding resolves this into the two equations (H,.^)2_^ = 0, g = 0 (1). The second of these gives y = ax-\-h, which satisfies the differential equation provided that h = ^J[l — a^). Thus the complete primitive is y = ax + V(l ~ ^^)' The first equation of (1) gives, on eliminating ~- by means of the differential equation, and this he terms the singular solution {singularis qucedam solutio prohlematis) , To apply to this example the general method, we find _xy±sj{x''-y'' + l) Hence, ~f = -^-— \x T ay X -^ 1 [ Introducing the condition -~ = cc , we shoirld apparently have the equations cc' + 1 = 0, hut of the second of these, as it does not involve y in its expression, no account is to be taken. The first making -^ infinite whether the upper or the lower sign be taken, and satisfying the differential equation, is a singular solution. Again, as also it is derived from the vanishing of a function under a negative index, it belongs to the envelope species. E I ART. 11.] EXAMPLES OP SINGULAR SOLUTIONS. 167 We may add that it might be found but less readily from the condition -^ f - ) = oo . The following example is intended to illustrate the use of the latter condition. Ex.2. Given t^ = CC-". ax Hence, since p = x~", the condition -f- = co cannot be satisfied. d The condition -^ ( — j = go gives and this is satisfied hy x = 0 if n be less tlian 1, but is not satisfied hy x = 0 i( 7i be equal to or greater than 1. Now the differential equation is satisfied hy x = 0, whatever positive value we give to 7i, as may be seen by expressing it dx in the form -j- = x". We conclude therefore that x = 0, is a singular solution of the proposed equation if 7i be positive and less than 1, but a particular integral if 7i be equal to or greater than 1. We infer too that the solution, when singular, be- longs to the envelope species. In verification, it may be observed that, if n be not equal to 1, the complete primitive is + c, ^ 1-n or ISTow if n is less than 1, the index in the second member is positive, and we cannot have x = 0 unless the quantity under 168 EXAMPLES OF SINGULAR SOLUTIONS. [CH. VIII. the index be made equal to 0. But this would give c = y. Hence, ic = 0 is a singular solution. If n be greater than 1, the index in the second member being negative we cannot have x — 0 unless the quantity under the index becomes infinite. But this it does if c is infinite. Here then a; = 0 is a particular integral. If n be equal to 1, the complete primitive is X — ce^, and this is reduced to cc = 0 by the assumption c = 0. Here then also ic = 0 is a particular integral. The following example is intended to illustrate a class of problems in which -^ admits of both a finite and an infinite value. Ex. 3. Given f - Ixifp + 4?/^ = 0. Here we find :p = xy^±^l{x'y-4.y^) (1). Therefore %_ J: dy~ 2y^ r "^ V(^'-4?/^) ^+-^] i% and this apparently becomes infinite when y = 0, and when x^ - Ay- = 0, i. e. for 2/ = 0, 2/=^. Let us inquire what are the true values of -J- . 1st. If y = — , we find, on substitution and reduction, dy x'\ ART. 11.] EXAMPLES OF SINGULAR SOLUTIONS. 169 which becomes infinite wliichsoever sign be taken. Hence, y = — is a singular solution ; and, from the mode of its origin, it is of the envelope species. 2ndly. If ?/ = 0, the value of -j- in (2) becomes infinite if the upper sign be taken, but assumes the ambiguous form - if the lower sign be taken. To determine its true value, we may expand the fraction ^-j- in ascending powers of ?/^. We thus find V(^'-4r) -—= — 'Ax -V [x ^ + &c. (^y 2?/M V ^ which, as before, gives -^ = go when, taking the upper sign, we make y = 0, but on taking the lower sign gives dy ixf V a; I 2 = - + terms containing positive powers of y. X 2 And this expression, on making ?/ = 0, assumes the value - . These results lead us to infer that the solution^ ?/ = 0, originates in two distinct ways from the primitive, which is in this case y= c^ {x- cf. It is evident that this is reduced to y^O, by either of the assumptions c = 0 and c = x. Hence the solution ?/ = 0 is a particular integral. At the same time it is to be noted that this solution pos- sesses all the geometrical properties of a singular solution. The complete primitive represents an infinite system of para- bolas whose axes are parallel to the axis of?/, — whose vertices all touch the axis of x, which thus constitutes a branch of their complete envelope, — and of whose parameters cach^ is inversely as the square of the distance of the corresponding 170 EXAMPLES OF SIXGULAPw SOLUTIONS. [ciI. VIIT. vertex from the origin of co-ordinates. Tlie nearer any par- ticular vertex is to tlie origin, the more does the curve to which it belongs approach to a straight line ; and the curve, if we may continue thus to speak, whose vertex is at the origin coincides Avitli the axis of x which is the envelope of the series. It might in a certain real sense be said that the particular and the general are here united. The following example shews, though by no means in the most extreme case, how slight may be the difference between a singular solution and a particular integral. Ex. 4. Given x-~ = y (}ogx + logy — 1), ax Representing -^- by p, we have ?/ (log X + log ?/ — 1) ^^'' '' ^ ' therefore ^ijo^^ + h^y ay X and tliis becomes infinite, 1st, if j/ = 0, 2ndly, if y — co , 3rdly, if a; = 0. The first only of these satisfies the differential equation, the assumption y = 0 reducing the indeterminate function y log?/ in the second member to 0 (Todhunter's Differential Calculus, Art. 158). We conclude, that ?/ = 0 is a singular solution, but from the nature of its origin not of the envelope species. Now the complete primitive is y = — , and, judging from this, it might at first sight seem as if ?/ = 0 were a particular integral corresponding to c = — co . We remark however that the primitive is not reduced to y = 0, by the assumption c = - CO , unless x he positive. If ic is negative we must make c = + CO to effect that reduction. In fact, the value of c which reduces the complete primitive to the form 3/ = 0, though in- dependent of x in all other respects, is dependent upon x for ART. 11.] EXAMPLES OF SINGULAR SOLUTIONS. 171 its sign, wliicli must always be opposite to the sign of x. And this connexion, slight as it is, determines the character of the solution. The following example illu?;trates a mode of procedure which may be adopted wlien -j- presents itself in the am- biguous form -, while the differential equation cannot readily be solved with respect to />. Ex. 5. Given j)" - ixyp + 8?/^ = 0. Differentiating with respect to y and p^ we find dp _ Axp — 16y , . d~y ~ Yf - 4.xy ^ ^' Equating to 0 the denominator, we have p = ~~— , and, v3 substituting this value in the differential equation, we obtain a result resolvable into the following equations, viz. 2/ = ^^'. 1/ = ^ (2), either of which satisfies the differential equation. On substi- tution in (1), the former of these values of 3/ makes -j- infinite, and is evidently a singular solution. The latter value of y reduces -/- to the form - . dy 0 To determine the real value or values of -/- when ?/ = 0, we must obtain from the differential equation, regarded as a cubic with respect to p, the three expressions for that quantity in ascending powers of?/, substitute them in the second member of (1), and then after reduction makey = 0. It will somewhat simplify the process if we transform the expressions by assuming^ = 2(//^. "VVe shall have dp _ 2.T^-4?/^ ,. dy'zfy^-xy^ ^'^' 172 EXAMPLES OF SINGULAR SOLUTIONS. [CH. YIII. while the differential equation will become f-xt + y^ = 0 (4), which, expressed in the form X X gives, bj Lagrange's theorem, X X Substituting in (3), and retaining those terms only which contain the lowest power of y, we have dp _ -2^_2 ^y — xy\ ^ Such is the value of —- corresponding to the value of t which is given bj Lagrange's theorem. That value of t vanishes with y. Its other values do not vanish with ?/, but approach the limits + x"^ as y approaches to 0 ; for if in (4) we make y = 0, we find 0 and + x^- for the corresponding values of t. Kow if in (3) we make 3/ = 0, t= ± ^/x^ we have dp -/ =co. dy From these results combined we infer that ?/ = 0 is a par- ticular integral, possessing the geometrical characters of a singular solution. It originates in fact from the complete primitive y = c(x — cY, either by making c = 0 or c = x. And that primitive, like the primitive of Ex. 3, represents a system of parabolas enveloped by one of their own number. Setting out from the primitive we find d , d7/ 1 1 dx ° dc x — c X- 3c ART. 11.] EXAMPLES OF SINGULAR SOLUTIONS. 173 X This expression becomes infinite when c = - corresponding to o 4 the sino-ular solution ?/ = — - icl It becomes infinite when c = x. •&■ 27 2 and assumes the value - when c = 0, — these cases belonging X to the particular integral y = 0. All these determinations agree wdth those of -^ obtained from the difi'erential equation. The following is an example of a special geometrical pro- blem generalized. Ex. 6. Determine a curve such, that the area intercepted between its tangent and the rectangular co-ordinate axes shall be constant and equal to — . The supposed area is aright-angled triangle wliose base and perpendicular, being the intercepts cut off by the tangents from the co-ordinate axes, are expressed by a; — ^ , and y — xp respectively. We have therefore {2j-xp)\x-^ = a\ Proceeding In the usual way the singular solution will be found to be representing an hyperbola, while the complete primitive repre- sents the series of tangents by w^hose successive intersection the curve is generated. To generalize the above problem we might suppose ^func- tional relation given between the intercepts. The differential equation would assume the form y-xp=f[x-^- 174 HiSToracAL account [ch. viii. Its complete primitive would always be determinable bj the method of Art. 9, Chap. vii. Or. since x — ^ = - ^ , it is easily seen that the equation is reducible to Clairaut's form y-xp = ^{p). The singular solution may then be found either as in Chap, vii., or by the direct application of the condition dp Geometrical problems which are of a truly symmetrical character frequently admit of this kind of generalization. Memarhs on the foregoing theory. 12. As the theory of the tests of singular solutions which has been developed in this Chapter differs in many material respects from any that have been given before, it is proper to shew in what its peculiarity consists. To this end it will be necessary briefly to sketch the history of this portion of analysis. Leibnitz in 1694, Taylor in 1715 (see Ex. 1, Art. 11), and Clairaut in 1734, had in special problems, and Euler in 1756 had in a distinct memoir entitled Exposition de quelques Para- doxes du Calcul Integral, examined, more or less deeply, various questions connected with the singular solutions of differential equations. Taylor in particular had first recog- nised the distinctive character of such solutions as set forth in their definition. The problem of the deduction of the singular solution from the differential equation seems however to have been first considered in its general form by Laplace. The same problem was subsequently investigated in a different manner by Lagrange, and again in a still different way by Cauchy. The state of the theory up to the present time w^ill be adequately represented by a summary of the results to which these several investigations have led. 1st. Laplace [Memoires de V Acadenne des Sciences, 1772), employing the method of expansions, arrived at results which agree, so far as they go, with those of this Chapter. They AliT. 12.] OF SINGULAR SOLUTIONS. 175 apply only to the envelope species of solutions, and the demon- strations of them rest essentially on the hypothesis expressed in (6), Art. 7. Lagran,2,-e, with whom originated a more fundamental idea of the method of the inquiry, was led to the less exact criteria dp _ dp_ dy ^ dx [Ccdcul des Fonctions, Le9on3 xiv — xvii.) Cauchy, whose method was founded on the study of the cases of failure of certain processes for obtaining the complete primitive in the form of a series, was led to the conclusion that a singular solution must satisfy one of the two following conditions, viz. d^j _0 dp _ together with a certain further condition, the application of which depends upon a process of integration (^loiguo, Ccdcul, Vol. IL p. 435). Upon these results the following observations may be made, I St. Although Laplace recognised the necessity of employ- ing in certain cases the condition -y- ( — ) = co , for -y- = co , dx \p) ' dij subsequent writers who have employed his method seem to have invariably omitted this qualification. 2ndly. The supposed criterion -^^ = go , introduced by La- grange, and since very generally adopted, as the proper accom- paniment of y- = CO , is erroneous. If we should apply it to Ex. 2, Art. 11, viz. p=x~'\ we should be led to the conclusion that £c = 0 is a singular solution whenever n is positive. We have seen however, both from the application of the true test, and by verification from the complete primitive, that ic = 0 is a singular solution only when n is less than 1. 176 HTSTOEICAL ACCOUNT [CH. VIII. THe principle of Lagrange's method was the same as that adopted in the present Chapter, and consisted in expressing -J- and -j- as derived from the differential equation, by means of differential coefficients derived from the complete primitive before the elimination of c. The fallacy which vitiated his results consisted in assuming that these expressions become infinite In consequence of the appearance of a vanishing factor in their denominators [Calcul des Fonctions, pp. 229, 232). Moigno, the expositor of Cauchy's views, also quotes La- grange's method and results as presented by Caraffa, but without involving any essential variation {Calcul, Tom. ii. p. 719). Professor De Morgan, in perhaps the latest publi- cation on the subject, adopts Lagrange's results, expressing, however, only a qualified confidence in his method {Cam- bridge FMlosopMcal Transactions, Vol. IX. Pt. ii. " On some points of tlie Integral Calculus"). And he illustrates these results by geometrical considerations which are sufficient to shew that they contain at least a considerable element of truth. Nor should this be thought surprising. For it is plain that Lao-rano-e's condition -^ = go , and the true condition ^ ^ dx — - ( - J = CO , are equivalent, except when the singular solu- tion makes p assume one of the forms 0 and co . And such cases do exist. Perhaps the peculiar difficulty of tliis subject has consisted in the faint and shadowy character of the line by which truth and eiTor are separated. 13. Of Cauoliy's tests the first, viz. 4^ = q, may certainly be set aside. Whenever -,- assumes an ambiguous form its di/ true value or values must be determined. This is ilkistrated in some of the foregoing examples. Professor De Morgan's observations on this subject in the memoir above referred to, are deserving of attention. The final criterion, which is peculiar to Cauchy's theory, seems to be founded upon what we cannot but regard as an unauthorized position as to the meaning of ART. 14.] OF SINGULAR SOLUTIONS. 177 a singular solution. Thus y = 0, the solution deduced hj the criterion -^ = co from the differential equation jp — y logy, is regarded by Cauchy as a particular integral. Xow although when X is real the complete primitive logy = c6^ reduces to ^=0 by the assumption c = - co , it does not necessarily do so when X is imaginary. Thus, if ic = 7r V(- 1), we must make c = CO , in order to give ?/ = 0. Cauchy's rule seems indeed to have been designed, contrary to the general spirit of his own writings, to exclude the consideration of imaginary values. Properties of Singular Solutions. 14. Various properties of singular solutions of the envelope species have been demonstrated. Of these we shall notice the most important. 1st. An exact differential eqiiation does not admit of a sin- gular solution. Let the supposed equation be d(i>{x,y) ^ d(t>(x,y) dy^^ dx dy dx ^ ^' and let y=f{x) be a relation actually satisfying it and assumed to be singular. On this assumption the primitive (f){x,y)=c must, on substituting for y its value /(a*), determine c as a function of x and not a constant. Let F{x) be the value of c thus determined, then <^ (x, y) = F(x) whence #(^^ y) ^ # {^, y) ^ ^ dF{x) ,^. dx dy dx dx which contradicts (1), since — i — cannot be permanently equal to 0, unless F{x) is constant. 2ndly. It follows directly from the above that a singular solution of a differential equation of the first order and degree^ onahes its integrating factors infinite. For let the proposed equation be Mdx + Ndy=0 (3), B.D. E. 12 178 PEOPEIiTIES OF SINGULAR SOLUTIONS. [CH. YIII. and let yit be an integrating factor. Tlien lJL[Mdx-tNdy) = 0 (4), will be an exact differential equation. Hence, a singular solution of (3), while it makes the first member of that equation to vanish, will not make the first member of (4) to vanish. Now comparing these members, this can only be through its making fx infinite. Ex. The equation x + y ~ =^-^ \J[x^ -^y^ — a^) has for its singular solution x^ ■\-y'^= aJ^, An integrating factor is and this the singular solution evidently makes infinite. Mul- tiplying the equation by its integrating factor and transposing we have the exact differential equation dy '^ dx dy ^/{x' + y^-d') dx and this is not satisfied by x^-\-y'^ = a^, the singular solution of the unrestricted differential equation. Srdly. Eveji when we are unable to discover its integrating factor^ a differential equation may he so prepared as to cease to admit of a given singular solution of the envelope s])ecies» This proposition is due to Poisson, and the following demonstration, which is purposely given in order to illustrate the nature of the assumption usually employed in the theory of singular solutions, does not essentially differ from his. Let us represent the singular solution by u = 0, and trans- form the differential equation by assuming ic and x as variables in place of ?/ and,ic. Suppose the new equation reduced to the form P=f{^,y) (5), where p stands for -7- . ^ dx ART. 14.] PROPERTIES OF {SINGULAR SOLUTIONS. 179 This equation is either satisfied or not satisfied hj u = 0. If it is not satisfied, the preparation in question has already been effected. If it is satisfied, the second member f(x, u) contains some positive power of w as a factor. Assuming that it can be developed in ascending pos^Ve ye powers of u it becomes p = Au"-+ Bu^+ ... + &C. where A, B, C, &c. are functions of x. Kow, for a singular solution -/- = x . Hence w = 0 must du render Aaii''-' + B/Su^-' + &c. = cc . But this demands that there should exist at least one negative power of u in the above development ; therefore a— 1, whicli is the lowest index, must be negative; therefore a being already positive must fall between 0 and 1. Hence we are permitted to express the differential equation in the form p = Qu% where a is a positive fraction, and Q does not involve u either as a factor or as a divisor. Dividing by u'^, we have du ^ Id, or- -~u' — =Q. Now u = 0 makes u^-'^=0, since 1 —a is positive. Hence the first member of the above equation vanishes, wdiile tlie second, not containing u as a factor, does not vanish. In its present form then the equation is no longer satisfied by u = 0. AVe see also that the property of being satisfied by u = 0 has been lost in consequence of a transformation wliich, exhibiting the singuUir solution in the form of a distinct alge- braic factor of ihe equation, permitted its rejection. Sec Art. 1 . 12—2 180 PROPERTIES OF SINGULAR SOLUTIONS. [CH. VIII. It has been shewn in the remarks on Clairaut's equation how, in the process of ascending bj differentiation to an equation of a higher order, a somewhat analogous effect is produced, the singular solution seeming to drop aside under changed conditions. 4thly. Lagrange has noticed that a singular solution will generally mahe the value of-r^^, as deduced from the differen- tial equation, assume the anibiguous form — . His demonstra- tion, in the statement of which we shall endeavour to exhibit distinctly the assumptions which it really involves, is sub- stantially as follows. Let the differential equation expressed in a rational and integral form be F{x,g,2y)=0 (1), then differentiating dF^ dF , dF ^ ^ ^^^ + ^^^ + ^^^ = ^ (2)- -p. dp dF dF ,^. Hence -/- = — ^- -^ -_ = co (3). ay ay d]} Xow F being rational and integral, -^ and -y- are so also, and therefore the above can only become infinite for finite dW values of a?, y, and ^j>, by supposing -— = 0. This reduces (2) to the form dF ^ , dF ^ ^ ,,, _^^+^^^^ = 0 (4). Xow, as obtained from the differential equation, dh/ _ dp dp dy dx^ dx 2y dx dF dFdy_ __ dx dy dx ~ — iP — ' dp ART. 15.] PROPERTIES OF SINGULAR SOLUTIONS. 181 an expression which the previous results reduce to the form - . We may remark that the condition -,^ = oo does not involve ay as a consequence dp=co in (2), so as to affect the legitimacy of the deduction of (4). For -^ = oo expresses a conditional proposition, whose antecedent is : li x be constant. Now in the deduction of (4) x is not supposed to be constant. Lagrange's demonstration is certainly only applicable to the envelope species of singular solutions. Of such solutions it expresses however an interesting property. For the dif- ferential equation being geometrically common both to tlie locus of the singular solution and to the locus of each parti- cular primitive, the ambiguity of value of y-g at the point of contact shews that that contact is not generally of the second order. In like manner, F{x, y, p) still being supposed rational and integral, the equation dF{x,y,p)_ dp -^ ^^^' shews by the theory of equations that the existence of a singular solution implies in general the existence of a series of points for which two values of ~ , usually different, come to agree, viz. the values of -^ in any particular primitive, and in the singular solution. O' 15. Mr De Morgan has made the very interesting remark, that when the condition -/- = go , or -^ ( i^^ strictness -^ — 1 =00 > dy dx\ dxpj does not lead to a solution of the differential equation, what it does lead to is the equation of a curve which constitutes the locus of points of infinite curvature (most commonly cusps) 182 EXERCISES. [CH. VIII. in the system of curves represented by tlie complete primitive [Transactions of the Cambridge Philosophical Society, Vol. IX. Part II.). Geometrical illustrations will be found in the memoir referred to. EXEECISES. 1. The complete primitive of a differential equation is y-\- c = ^/ [x^ + y^ — a') , where c is the arbitrary constant. Shew that the singular solution is x^ -\- y"^ = a^, and that it may be connected with the primitive by either of the equivalent rela- tions c = —y and c = ^J[a^ — x^). 2. Why is the above singular solution deducible by the dx dv application of either of the conditions -7- = 0, -^ = 0? 3. Expressing the primitive in Ex. 1 in a rational and integral iorm (^ [x, ?/, c) = 0, deduce the singular solution by the application of the condition -~ = 0. 4r. The complete primitive of a differential equation being x — a = {y — cy\ shew that the singular solution is deducible dx by the application of the condition -^ = 0 but not by that of the condition -,- = 0, and explain the circumstance. 5. The differential equation, whose complete primitive is given in Ex. 1, may be exhibited in tlie form [x"^ — a^) p^ — 2xyp — a?^ = 0. Hence also deduce its singular solution and thereby verify the previous result. 6. Form the differential equation whose complete primitive is given in Ex. 4, and shew that the singular solution is de- ducible by the application of the condition -y- - = co but not CH. VIII.] EXERCISES. 183 by that of the condition -j-=^, and explain tliis circum- stance. 7. Sliew that tlie sinsruhar solutions in the last two ex- amples are of the envelope species. 8. The differential equation y = px -\- — (Ex. 2, Art. 5) lias y = cx-\-— for its complete primitive, and y"^ = 4.mx for its c singular solution. Verify in this example the fundamental relation -/- = -^ \o^-^. ay ax ac 9. Deduce both the singular solution and the complete primitive of the differential equation y =px + \/ {TJ^ + d^p^), and interpret each, as well as the connexion of the two, geometri- cally. 10. The following differential equations admit of singular solutions of the envelope species. Deduce them. xY -2 [xy -'i) p -\- y"" =^0y {y — xjy) [mp — n) = mnp, y = {x-l)p-p\ 11. The equation (1 - x^) p + xy -a = 0 is satisfied by the equation y = ax. Is this a singular solution or a particular integral ? 12. The equation y = ^ is satisfied by y = 0, which also makes -^ (-j = co. Nevertheless y = 0 is a particular inte- gral ■ Shew that this conclusion is in accordance with the general Theorem (Art. 11). IP). The equation p [x^ — I) = Ixy \o£^ y has a singular solution which is not of the envelope species. Determine it. 184 EXERCISES. [CH. VIII. 14. Determine also the complete primitive in the last ex- ample, and shew how the singular solution arises. 15. The equation {p-yY- 2^'y {p-y) = ^xY - 4a?y log y is satisfied by 3/ = 0. Shew that this is a singular solution hut not of the envelope species. 16. Find singular solutions of each of the following equa- tions, and determine whether or not they are of the envelope species. 1. p^+'22yx^=Ax^y. 2. xj)'^ — 2yp 4- 4ic = 0. 3. xjy = n (a;" + {y - x"") log {y - a;")}. Geometrical Applications. • In solving the following problems, the diiferential equation being formed, its complete primitive as well as its singular solution is to be found and interpreted. 17. Determine a curve such that the sum of the intercepts made by the tangent on the axes of co-ordinates shall be constant and equal to a. 18. Determine a curve such that the portion of its tangent intercepted between the axes of x and y shall be constant and equal to a. 19. Find a curve always touched by the same diameter of a circle rolling along a straight line. 20. Find a curve such that the product of the perpendicu- lars from two fixed points upon a tangent shall be constant. (Euler. See Lagrange, Gale, des Fonctions, p. 282.) CH. VIII.] EXERCISES. 185 (Representing the product by k^, and the distance between the given points by 2m^ making the axis of x coincide with the straight line joining them and taking for the origin of co-ordinates the middle point, the differential equation is [y-{x-\-m)p] [y-[x-'m)p] ^^^^ Its singular solution is 21. Deduce also the complete primitive of the above dif- ferential equation, 22. If the primitive of a differential equation be expressed in the form <^ (ic, y, a) — 0, the condition ;t^ = 0 may be ex- pressed in the form d^y^ ^d^y^ ^ ^_ ^^^_ ^_ da ay Hence it has sometimes been laid down that ! ' ' = co ay will lead to a singular solution. Raabe, in Crelles Journal {TJeber singiddre integrale, Tom. 48), points out that this rule may fail if at the same time ' ' — - should become in- finite. Can it fail in any other case ? 23. Exemplify Raabe's observation in the equation x + c-^ {6cy - 3c') = 0, which is the complete primitive of Sxp"^ — Qyp -}- x + 2y = 0. At the same time shew that the singular solutions are y — x = 0 and 3?/ + ic = 0. ( Crelle, lb.) 24. The complete primitive of a differential equation is (c- X + yY - ^ {x + y) {c-x + yY +\ =0, 186 EXERCISES. [CH. VIII. Eepresenting its first member, which is rational and integral, hy (j), the condition -j^ = 0 assumes the form 3 (c - a? + ?/) (c - 3aj - ?/) = 0. Shew that c — a? 4- ?/ = 0 will not lead to a solution of the differential equation at all, while c — 3x — i/ = 0 will, and explain this circumstance hj a reference to Art. 4. Note. The reader is reminded that in all references to the general condi- {x) in its successive states of increase are known. Thus since A(/) {x) = {x)} = [(f){x + 2Aa:) - (^ (a; + Ax)] - {(f) (x + Ax) - cj) {x)] = {x), &c. are given, tlie successive values of the function (j){x), viz. the values (j){x-\-Ax), ^{x-\-2Ax), &c., are thereby made deter- minate. Geometrically we may represent (x) by ?/, the ordi- nate of a curve, or of a series of points in the plane x, y^ and therefore functionally connected with the abscissa x, !N'ow the view to which reference has been made is that which, 1st, presents the differential equation (1) as the limiting form of the relation expressed by the equation Aic«~-^r'^' A^' Ax'' '" Ax'^-'j ^ ^' A^ approaching to 0; 2ndly, constructs the latter equation in geometry (the arithmetical or purely quantitative construc- tion being therein implied) by a series of points on a plane, of which the first n, viz. those which answer to the co-ordinates X, x-\- Ax,.. . x-\-{n—\) Ax, have the corresponding values of y arbitrary, while for .all the rest the values of y are deter- mined; 3dly, represents the solution of the differential equa- tion as the curve which the above series of points in their limiting state tend to form. According to this view, the n arbi- trary points in the constructed solution of the equation of differences (2) give rise to one arbitrary point in the limiting curve, accompanied hj n — 1 arbitrary values for the first n — \ differential coefficients of its ordinate. And this mode of consideration appears the simplest, because it assumes no more than the definition itself demands of us when we attempt to realize the geometrical meaning of a differential coefficient as a limit. We may however add that when by the consi- deration of the limit, the mere existence of a primitive has been established, other considerations would suffice to shew that in its complete form it will involve n arbitrary constants and no more. The fact that each integration introduces a single constant is a direct indication of the fact. An indirect proof of a more formal character will be found in a memoir by Professor De Morgan [Transactions of the Cambridge Fhi- losopMcal Society, Vol. IX. Pt. II.). ART. 2.] AN ORDER HIGHER THAN THE FIRST. 189 The atove theoiy may be illustrated by the form in which Taylor's Theorem enables us to present the solution of a differential equation of th.Q'n}'^ order, as will be seen in the following Article. Solution hy development in a series. 2. Reducing the proposed equation to the form dx- -^V''^' dx' '"dx'^-'J ^^^' and differentiating with respect to x, the first member becomes J „;[, while the second member will in general involve all the differential coefficients of y up to -y-— . If for the last we substitute its value given in (3), the equation will assume the form dx'^^' J^^y^ dx'"' dx^^-'j ^^^• Thus , ,^;( is expressible in the same manner as —{ , viz. in terms of x, y, and the first 7i — l differential coefficients of y. Differentiating (4) and again reducing tlic second member by means of (3) we have a result of the form ^^n.. j^\^,y, ^^, -"dx'^-'] ^^^' and in this form and by the same method all succeeding dif- ferential coefficients may be expressed. Hence reasoning as in Chap. II. Art 12, we see that sup- posing y to be developed in a series of ascending powers of x — x^, where a:„ is an assumed arbitrary value of.r, the co- efficients of the higher powers of ^ — ^^ beginning with {x — x^Y will have a determinate connexion, established by means of the differential equation, with the coefficients of the inferior powers of x — x^^. The latter coefficients, n in number, 190 SOLUTION BY DEVELOPMENT IN A SERIES. [CH. IX. "beginning with the constant term which corresponds to the index 0, and ending with — -r -r^ , which is the coefficient of [x — icJ^'S will be perfectly arbitrary in value. To exhibit the actual form of the development let y^, 7/^y... 2/n-i be the arbitrary values assigned to y, -^^ , ... - . „:^ when X = x^. Also let /, /^, f^, &c. represent the values which the second members of the series of equations (3), (4), (5) assume when we make in them ic = x^; then In this expression the arbitrary values of j/ and its w — 1 first differential coefficients corresponding to an assumed and definite value of x, viz. t/^, Vv'-Vn-x ^i'^ the n arbitrary con- stants of the solution, the values oi f^, fi^^^, &c., being deter- minate functions of these, and therefore not involving any arbitrary element. Any function of arbitrary constants is itself an arbitrary constant, and thus it may be that an equation has effectively a smaller number of arbitrary constants than it appears to have from the mere enumeration of its symbols. As a general prin- ciple we may affirm, that the number of effective arbitrary constants in the solution of a differential equation while on the one hand equal to the index of the order of the equation, is on the other hand to be measured by the number of conditions which they enable us to satisfy. Systems of conditions to be thus satisfied will indeed vary in form, but there is one system which we may consider as normal and to which all other systems are in fact reducible. It is that which is de- scribed above, and which demands that to a given value of x a given set of simultaneous values of y and of its differential coefficients up to an order less by 1 than the order of tlie equation shall correspond. Conversely, the arbitrary constants ART. 2.] SOLUTION BY DEVELOPMENT IN A SERIES. 191 of a solution may be said to be normal, when they actually represent a simultaneous system of values of y and its succes- sive diflferential coefficients up to the number required. Ex. Given -^ = -j-^-y^* Kequired an expression for ij in the form of a series such that when a; = 0, v and , shall •^ ax assume the respective values c and c'. Differentiating, we have dx^ dj? ^ dx ^aJ^^^'^'^dx'^ ^^^^^ equation, by similar reduction, and so on. Hence, corresponding to .r=0, we have the series of values, dy , d'^y , ., g = c^+(l + 2c)c', dx and so on. Hence, ?/ = c + c a; + — - — ;k? ^^=c^+2c^+(l + 4c)c'+2c^ c^+(l + 2c)c' 3 c'+2c'+(l + 4c)r' + 2c'-^ , , 192 FINITELY INTEGRABLE FORMS. [CH. IX. Finitely Integrahle Forms. 3. As tlie difficulty of the finite integi'ation of difierential equations increases as their order is more elevated, it becomes important to classify the chief cases in which that difficulty- has been overcome. It will be found that for the most part these cases are characterized by some one or more of the following marks, viz. 1st, Linearity, the coefficients being* at the same time either constant or subject to some restriction as to form ; 2ndly, Absence of one or more of the variables or their differ- ential coefficients; 3rdly, Homogeneity; 4thly, Expressibility in the form of an exact differential or in a form easily re- ducible thereto by means of a multiplier. The subject of linear equations being of primary importance, we shall devote the remainder of this Chapter to its discussion. But as it will be resumed in another part of this work, and in connexion with a higher method, we propose to notice here only the more important general properties of linear equations, and to illustrate them in the solution of equations with con- stant coefficients. Linear Equations. 4. The type of a linear differential equation of the n^^ order, (Chap. I. Art. 4), is ^^ + A.^{ + A,^...+X„^ = X (7), in which the coefficients X^^X,^...Xn and the second member X are either constant quantities or functions of the independent variable x. Considering, first, the case in which the second member is 0, the following important proposition may be established. Prop. If ?/j , ?/2 , . . . ?/„ represent n distinct values of y, which individually satisty the linear equation, ART. 5.] LINEAR EQUATIONS. 193 then will the complete value of y be Cj, c.^,.--c„ being arbitrary constants. In other words the com- plete value of y is the sum of n distinct particular values of y, each containing an arbitrary constant. For on substitution of the assumed general value of y in (8), we have a result which may be arranged in the follow- ing form, viz. dx'' + ^.^:^l + ^.jnM'- + X.i/. cU + c, ;>=o...(9). dy. .. 6?" ..d" j_n , Y -^^b . X ^ '^'^ + Y V Now each line in the left-hand member of the above equa- tion is, from the hypothesis as to the values of y^, ?/., ?/„, equal to 0. Hence the equation (9) reduces to an identity, and the theorem is established. The problem of the complete solution of a linear equation of the n^^ order whose second member is equal to 0 is, there- fore, reduced to that of finding n distinct particular solutions, each involving an arbitrary constant. 5. Prop. To solve the linear equation with constant coefficients when the second member is 0. "Were the proposed equation of the first order and of the form its solution would be y = ce B. D. E. 13 194 LINEAR EQUATIONS. [ciI. IX. From tills result, and from the known constancy of form of the differential coefficients of exponentials, we are led to examine the effect of such a substitution in the equation Assuming then y = Ce'"'', and observing that d" [Ce'"''') _n/imx dx'' we have, on rejection of the common factor Ce'"'', the equation m'' + a^m''~' + a^rjr\.. + a„=0 (11), the different roots of which determine the different values of m which make y = 6V^ a solution of the equation given. When those roots are real and unequal, we have, therefore, on representing them by ??ij, m.^, ... w„, the system of 7i par- ticular solutions, 7/ = c/n 7j= a/n ...:y= ae'""^ (12), from which by the foregoing theorem we may construct tlie general solution, y=Cy-^+a/^-^\.. + C^e^'-'- (13). The equation (11) by which the values of ??i are determined is usually called the auxiliary equation. Ex. Given ^,-3^ + 2?/ = 0. Here, assuming y = C'e'"^, we obtain as the auxiliary equation m"" - Sm +2 = 0. Whence the values of m are 1 and 2. The corresponding particular integrals are y = C^e", and y = (7^e^*, and the com- plete primitive is AliT. G.] LINEAR EQUATIOXS. 195 G. If among tlic roots, still supposed unequal, imaginary pairs present themselves, the above solution, though formally correct, needs transformation. Let a ±h\/ — 1 represent one of these pairs, then will the second member of (13) contain a corresponding pair of terms of the form which we may reduce as follows, = Ce""^ (cos hx-\-'J - i sin hx) + C'e'"" (cos hx - V^^ sin Ix) = (0+ C") e""cos 5^4- ((7- C) ^/(- 1) e^'^sin 6^, or, replacing C + C and {C — C) \/ {— 1) by new arbitrary constants A and B, Ae'^'coshx + Be^'smhx (14). Ex. Given -T7. - ^ ^- + 1-"Z/ = ^• Assuming 7/ = C'e'"'', the auxiliary equation is m""- 4,711 + 13 = 0, whence »i = 2 + 3 V (— !)• 1'l^e complete solution therefore is 1/ = ^e^" cos 3x + Be"'' sin 3^. 7. Lastly, let the auxiliary equation have equal roots whether real or imaginary, e.g. suppose m,^= m^. Then in the general solution (13) the terms (7^6"'!'^+ (7/'"-^ reduce to a single term ( C^ + CJ e*''i''', and the number of arbitrary con- stants is effectively diminished, since G^ + C^ is only equiva- lent to a single one. Here then the form (13) ceases to be general. To deduce the general solution when ;/?., = m^ let us begin by supposing m.^ to differ from 7??^ by a finite quantity /», and 13—2 196 LINEAR EQUATIONS. [CH. IX. examine the limit to which the terms of the solution, then really general, approach as h approaches to 0. Now = e-^^ [a -\-Bx + Bh ^^- + &c.) ; on replacing C^ + C^ and CJi bj A and B, new arbitrary con- stants. This change it is permitted to make, however small h may be, provided that it is not equal to 0. The limit to which the last member of the above equation approaches as h approaches to 0 is This then is the form which must replace (7^e'"''' + C./'-^ in the o:eneral solution. m,^, m^. Suppose next that there exist three equal roots w^, Then the terms C/'^''- + C/'-^ + C/'-^'^ being replaced by e^'^^^[A+Bx)-vC/'-% make m.^ = m^ + Iz, The above expression becomes ^-.-^A^Bx^C^i') = 6'"^^{^ + C3+ [B^ CJc)x+C,^^x^+ ^3Y;|7^^^ + &c. = 6'"^-^(^' + i?'^ + CV + -^iC^+&C.) (15), on making A+C, = A\ i? + C,h = B\ ^ = C, Here A\ B\ C being functions of the arbitrary constants A, B, C provided that k is not actually 0, may themselves be regarded as arbitrary constants. If we so consider them in ART. 8.] LINEAR EQUATIONS. 197 (15) and then make h tend to 0, we see that tlie limiting form of the expression is And in precisely the same way, were there r roots equal to 7?2j, we should have for the corresponding part of tlie com- plete value of ?/j the expression e'"^^(^, + ^2ic + ^3.x^..+^X"') (16). Thus the difference which the repetition of a particular root m^ produces is that the coefficient of the exponential e'"'"" is no longer an arbitrary constant, but a polynomial of the form A^-\- A^x -\-&Q,.^ the number of arbitrary constants involved being equal to the number of times that the supposed root presents itself. Ex. Give,4>-^-|^ + y = 0. dx dx^ dx ^ Here, assuming y = Ce"'"", the auxiliary equation is m^ — m^ — 771 + 1 = 0, the roots of which are — 1, 1, 1. Thus, corresponding to the root — 1, we have in y the term Ce''', while to the two roots 1, we have the term [A + Bx) e^. The complete primitive tliere- fore is y=Ce-''-\-[A + Bx)e\ 8. It follows from (16), that if a pair of imaginary roots a ±h^ —I present itself r times, the corresponding portion of the complete value of y will be ( c; + (7,0^. . . + c^^-') e'^^^'^ ^-v ( c; + c> . . . + c;x^') 6«'-^^ ^, which, substituting for e^'"^"^ and e"^""^'^ their trigonometrical values and finally making c, + o; = A„ (c. - c;) v^i = b, , &c., assumes the form {A^ + A^x . . . + A.x"'') e""" cos hx + {B^ + B^x...-\- B.x'--') e"^ sin hx. 198 LINEAE EQUATIONS. [CH. IX. Hence, therefore, the repetition of a pair of imaginary roots a±h^/—l changes also the two arbitrary constants of the ordinary real solution into polynomials, each of which involves a niimljsr of constants equal to the number of times that the imaginary pair presents itself. Ex. Give„3+2.'3 + «V=0. Assuming y = 6V"'', the auxiliary equation is nc" + 2?i'wi' + n'' = 0, whence m has two pairs of roots of the form ±- n \/(— 1). For one such pair the form of solution would be y — A cos nx + B sin nx. For the actual case it therefore is y = {A^ + A^x) cos nx + {B^ + B^x) sin nx, 9, The above, which is the ordinary method of investi- gating the form of the complete solution when the auxiliary equation involves equal roots, rests on the assumption that a law of continuity connects the form of solution when roots are equal with the form of solution when the roots are unequal. Now, though it is perfectly true that such a law does exist, its assumption without proof of that existence must be regarded as opposed to the requirements of a strict logic. In all legiti- mate applications of the Differential Calculus it is wdth a limit that we are directly concerned. Here it is with some- thing which exists, and Vv'hich admits of being determined in- dependently of the notion of a limit. 72 J Thus if wx take as an example -^-^ — 2-~-\-y = 0, in which CLX CtX the auxiliary equation 9?i^ — 2m + 1 = 0 shews that the values of w are each equal to 1, we are entitled to assume as a par- ticular solution ART. 10.] LIXEAll EQUATIONS. 199 Let us now substitute this value of?/ in the given equation regarding C as variable, and inquire whether it admits of any- more general determination than it has received above. On substitution we find simply wlicnee C=A-\-Bx, Thus while the correctness of the solution furnished by the assumption of continuity is esta- blished, it is made manifest that this assumption is not in- dispensable. We sliall endeavour to establisli upon other grounds tlie theory of these cases of failure in a future Chapter. Mean- while it may be desirable to shew that the form (IG) actually satisfies the differential equation when r values of m are equal. In the given equation assume y = Ce"V, s being an integer less than r. Irom the theorem tor -j—^ it easily follows that the result will be of the form f{,n) X' +/■ (m) sx'-^ +/" (m) ''-^^^^~+ ■ • • +/"'(«') in whic]i/(7?2) represents the first member of (11). But that equation having by hypothesis r equal roots, we know by the theory of equations tliat /(m)=0, /W=0,... /"W=0, are simultaneously true. Thus the differential equation is satisfied. And being satisfied for tlie particular value of?/ in question it is satisfied by (IG), which is the sum of all such values. 10. The results of the previous investigation may be sum- med up in the following rule. 200 LINEAR EQUATIONS. [CH. IX. KuLE. The coefficients heing constant and the second mem- her 0, form an auxiliary equation hy assuming y = Ce'""*, and determine the values of m. Then the complete value of y will he expressed hy a series of terms characterized as follows, viz. For each real distinct value of m there icill exist a term Ce^ ; for each pair of imaginary values a±h \l[—\), a term Ae""^ cos hx + Be'"'' sin hx; each of the coefficietits A, B, C being an arbitrary constant if the corresponding root occur only once, hut a polynomial of the (r—iy^ degree loith arhitrary constant coefficients, if the root occur r times. Lx. Given ^5-t4-2:t4 + 2t^ = 0. dx' dx dx^ dx Here the auxiliaiy equation is m^ — m^ — 2m^ -\-'2m = Q, wlience it will be found that the values of m are 0, 1, 1, -i±v'(-i). The complete primitive therefore is y=C+{C^+C^x)e'+ 6;e-" cos x + C.e"" sin x. 11, To solve the linear equation with constant coefficients when its second member is not equal to 0. The usual mode of solution is 1st to determine the com- ])lete value of y on the liypothesis that the second member is 0; 2ndl7, to substitute its expression in the given equation regarding the arbitrary constants as variable parameters; 3rdly, to determine those parameters so as to satisfy the equa- tion given. Supposing the given equation to be of the w^^ degree, n ■[parameters will be employed. These may evidently be sub- jected to any n—\ arbitrary conditions. Now that system of conditions which renders the discovery of the remaining rela- tion (involved in the condition that the given differential ART. 11.] LINEAR EQUATIONS. 201 equation shall be satisfied) the most easy, is that whicli demands that the formal expression of the n—l differential coefficients shall, like the formal expression of y, be the same in the sys- tem in which c^, Cg,...^,, represent variable parameters, as in the system in which they represent arbitrary constants. The above method is commonly called the method of tlie variation of parameters. It is, as we shall hereafter see, far from being the easiest mode of solving the class of equations under consideration; but it is interesting as being probably the first general method discovered, and still more so from its containing an application of a principle successfully em- ployed in higher problems. Ex. Given -~ + n^y = cos ax. Were the second member 0, the solution would be y = c^QO^nx -^ c^^uinx (a) . Assume this then to be the form of the solution of the equa- tion given, Cj, c^ being variable parameters, but such that -^ shall also retain the same form as if they were constant, viz. dy . ,,v "T- = — c,n ^innx + cjico&nx [o). dx ^ ^ ^ Now the unconditional value of -j^ derived from (a) is ctx dy . _ dc, . dc, J- — — c.n sm nx + cji cos nx + cos nx -r^ + sin nx -r^ , ax ^ 2 ^^^ ^^ which reduces to the foregoing form if we assume dc, . dc^ ^ , . cos nx -~-\- sm nx -r^ = 0 (c). ax dx This then is the condition which must accompany {a). 202 LINEAR EQUATIONS. [CH. IX. Xow differentiating [l) and regarding c^, c^ as variable, we have ^'y 2 2 . . dc, dc^ -r^, — - c.n cos 7ix — cji sm ?2ic — n sni «a? -^^ + n cos ^la; -v-^ . 3 above values of y an equation, we have Substituting the above values of y and -y4^ in the given dc. , ^c^ , ,. — 7i sm nx~ + ?i cos ?ia7 y^ = cosa^ (a), and this equation, in combination with (c), gives dc. 1 . Jc., 1 -7-^ = cos ax sm na?, -r- = - cOs ax cos nx, dx ' n dx n y 1 fcos {n-{- a) x cos fw — a)x) ^ whence c, = - ^ ^— — ^ + ^ >—\ + C,, ^ 2n [ n-\- a n — a ) ^ 1 [sin {n -\- a) X sin [n — a)x ^ 2)1 [ n + a n — a J ^ Lastly, substituting these values in [a) and reducing, we have cos ax ^ „ . , . y = -7, 7, + C, cos7i:r + C^?>\nnx ie). ^ n ~ a" ^ ^ This solution fails if n = a. But giving to (e) the form cos ax — cos nx ^., ^ . y = 5 ^ {- C' cos nx + C, sm ??x, "^ n —a ^ and regarding the lirst term as a vanishing fraction when « = a, we find X sm ?2a:^ „ , ^ . y = — 1- C' cos nx + G' sm 7ia?. •^ 2n ^ Or we might proceed thus. Differentiating twice the equation d'y . -TT + n~y = cos ?za7, "^'^'Z . .2 ^V _ ,2 we <]ret -7- , + ?r -7^, = — n cos ??a:, ° dx dx AKT. 12.] LINEAll EQUATIONS. 203 Ilcnce eliminating cos nx an equation wliosc complete solution is • y = {A 4- Bx) cos nx -{- {C + Dx) sin nx. Substituting tliis in tlic given equation we find i?=0, D = — , whence 2/1 y = AQO?>nx-\- iC ■{•---] smnxy AvliicK agrees with the previous solution. The latter method, which is general, consists in forming a new equation of a higher order, but with its second member free from that term which is the cause of failure. As by the elevation of the order of the equation superfluous constants are introduced, the relations which connect them must be found by substitution of the result in the given equation. 12. To the class of linear equations with constant coeffi- cients all equations of the form («+5^)«g+^(a+J^)»--^>+i?(« + fo)-gl^..+i3/ = A-, A, B,...L being constant and X a function of x, may be reduced. It suffices to change the independent variable by assuming a + hx = e^ Ex. Given {a + Ixf -^., + h{a-\- hx) -j- + ?ry = 0. Assuming a-{-hx = e\ we find dx~~^^ dt' dx'~ We dtl' 204 LINEAR EQUATIONS. [CH. IX. Hence, hy substitution in the given equation, we have the solution of which is y—(jcosj--\- C sm J- , in which it only remains to substitute for t its value log [a-\-hx), 13. Beside the properties upon which the above methods are founded, linear equations possess many others, of which we shall notice the most important. We suppose, as before, y to be the dependent, x the independent variable. 1st. The complete value of ?/ when the linear equation has a second member X will be found by adding to any particular value of?/ that complementary function which would express its complete value were the second member 0. Representing the linear equation in the form (7), let ?/j be the particular value of y which satisfies it, Y the complete value which would satisfy it were the second member 0; and assume y=y^-\- F. The equation then becomes and this becomes an identity, the first line of its left-hand member being by hypothesis equal to X, and the second line equal to 0. Ex. Thus a particular integral of the equation being y= :^— , its complete integral is a AKT. 13.] LINEAR EQUATIONS. 205 The above property, which relates to tlie generaJizwrj of a particuhir solution, is important, because, as we shall hereafter see, a particular solution of a linear equation may often be obtained by a symbolical process which does not involve even the labour of an integration. 2ndly. The order of a linear differential equation may always be depressed by unity if we know a particular A'alue of y which would satisfy the equation were its second member equal to 0. It will suffice to demonstrate this property for the equation of the second order 3 + A'.| + X^ = A- (18). Let y^ be a particular value of y when A'=0, and assume y = y^v. Substituting, we have S + A'.t. + A>,)„ dx^ " ^ dx the first line of which is by hypothesis 0. In the reduced equation let --- = u, then we have 2'.£+(^S+'^>>'=^^' (^^)' a linear equation of the first order for determining u. And this being found, we have V = I iidx + c. J In the particular case in which X= 0, we find from (10) tc = whence y = ^j^(^C\ —^dx+ Cj (20). 206 LINEAK EQUATIONS. [CH. IX. 3rdly. Linear equations are connected by remarkable ana- logies with ordinary algebraic equations. This subject has been investigated chiefly by Libri and Liouville, who have shewn that most of the characteristic properties of algebraic equations have their analogues in linear differential equations. Thus an algebraic equation can be deprived of its 2nd, 3rd, . . . 7'^^ term by the solution of an algebraic equation of the 1st, 2nd, ...(/• — l)*^ degree. A linear differential equation can be deprived of its 2nd, 3rd, ...r*^ term by the solution of another linear differential equation of the Ist, 2nd, ... (r — 1)^*" order. This may be proved by assuming ?/ = ?'^i, and properly de- termining V so as to make in the resulting equation y^ assume the required form. Again, as from two simultaneous algebraic equations, we can by the process for greatest common measure obtain a de- pressed equation satisfied only by their common roots, so from two simultaneous linear differential equations we can by a formally equivalent process deduce a new equation of a de- pressed order satisfied only by their common integrals. This is best illustrated by example. Ex. Required the common integrals, if any, of the equations d'^y ^ dy ddu" dx Differentiating the second equation and then eliminating -^'(^ and -r4 , we find the depressed equation dx^ dx"' If we differentiate this we shall find that the result is merely an algebraic consequence of the two equations last CII. IX.] EXEUCISES. 207 written, not an algebraically new equation. Thus tlic process of reduction cannot be repeated. We have therefore as the only common integral. EXEECISES. 1. ^^-7^ + 12^ = 0. dx^ dx ^ d'^y „dy 3. Integrate '^^ - 4 '^'3 + 6 ?| - 4 ^?^ + ^ = 0. * dx'' dx^ dx^ dx -^ 5. y^-3^ + 4?/=0, it being given that one of the roots of the auxiliary equation, in — oiu' -\- 4 =0, is — 1. C). -y^l - 2 , -'( + 2 vi - 2 y + ^ = 1 . dx dx dx dx ^' dx} "'"^dx^ -^ 8. What form does the solution of the above equation assume when h =1? ^- "^ dx^ ""d^-^'^' -I ri / \ " d^V . f \ dy , 10. {x + ci)-^-^(x + a)£-^C?/ = 0. 1 1. Intecrate ^ " 2^^ ^ + h'j'^ = 0. 208 EXERCISES. [CH. TX. 72 7 12. A particular integral of (1—^0 t^z~^~j: ~^^y = ^ h y= (7e"°'"~^^, find the complete integral by the method of Art. 13. 13. The form of the general integral might in the above case be inferred from that of the particular one without em- ploying the method of Art. 13. Prove this. 14. It being given that . / . cos x\ r^f sin X ;i/ = A[smx-h — — 1 +i) cos £c ~ d"^!! / 2 \ is the complete integral of the equation yK, + [ 1 ^j ?/ = 0, find the general integral of -~ + ( 1 2 ) 3/ = ^^• clx \ X / 15. Explain on what grounds it is asserted that the com- plete integral of a differential equation of the 11^^ order contains Qi arbitrary constants and no more. 16. Mention any circumstances under which it may be advantageous to form, from a proposed differential equation, one of a higher order. In deducing from the solution of the latter that of the former, what kind of limitation must be introduced? ( 209 ) CHAPTEH X. EQUATIONS OF AN ORDER HIGHER THAN THE FIRST, CONTINUED. 1. 'We have next to consider certain forms of non-linear equations. Of the following principle frequent use will be made, viz. When either of the i:)rimitive variables is icanting, the order of the equation may he depressed hy assuming as a dependent vari- able the lowest differential coefficient which presents itself in the equation. Thus if the equation be of the form and we assume ^(-i-S)=« «= I- (^)' we have, on substitution, the differential equation of the first order, ^(^'^'£)=« (^)- If, by the integration of this equation, z can be determined as a function of x involving an arbitrary constant c, {suppose s = 0 (a;, c)}, we have from (2) "whence integrating y=\(f>{x, c)dx-\- c. B.D.E. 14 210 EQUATIONS OF AN ORDER HIGHER [CH. X. If the lowest differential coefficient of y which presents itself be of the second order, the order of the equation can be depressed by 2, and so on. A similar reduction may be effected when x is wanting. Thus, if in the equation of the second order 4'!' 3)- W' we assume -r = ;?, we have ax '■ d'^y _ dp _ dv dy dp dx^ dx dy dx ^ dy ' by means of which (4) becomes -^(^'^■^1') = ' ('^- Sliould we succeed by the integration of this equation of the first order in determining p as a function of y and c, sup- dii pose p = cf>{y, c), the equation ~ =;:>, will give dx= ^y v» hence fv/'S + c (6). Ex. Suppose 1+g) +2, || = 0. Put -J- —v\ thus dx -^ therefove f+i^^ = 0> therefore log 3/ 4- log V (1 +i^") = constant, ART. 2.] THAN THE FIRST, CONTINUED. therefore y^ (1 ■\-]f) = constant = h, therefore '-"•-"r therefore dx y dy-^{¥-yr therefore x=-sl[V-f)+a, where a is a constant : } thus fiualljT, y ,2+(x-a)^ = J% 211 2. In close connexion with the above proposition, stand the tliree following important cases. Case I. When hut one differential coefficient as Avell as but one of the primitive variables presents itself in the given equation. 1st. Let the equation be of the form -^ = -X", we have by successive integrations dx and finally y ^,=jjxdx' + cx + c\ = jj ...Xdx''+c^x''-'+c^x'^\..+ c, (7 We shall hereafter shew that the first term in the second member may be replaced by a series of n single integrals. 2dly, If the equation be of the form -^^ = Y, it is not gene- rally integrable, but it is so in the case of n = 2. Thus there being given dx^^ ' U— 2 212 EQUATIONS OF AN ORDER HIGHER [CH. X. we have dx dx^ dx ' and integrating Hence {2fYdy+G)i' {2jrdy+q'- As a particular example, let -^ = a^i/, \ Case II. AVhen tlie given equation merely expresses a (relation between two consecutive differential coefficients. i Suppose the equation reduced to the form dx''~J\dx'-') ^'^^' then, assuming , ,J^ = ^, we have dz whence dx = f(^)' \m'^ (^«^: "' '/(-) ART. 2.] THAN THE FIRST, CONTINUED. 213 If, after effecting tlie integration, we can express z in terms of X and c, suppose z = (f> {x, c) we have finally to integrate ^^~\ = {x,c) (11), which belongs to Case i. But if, after effecting the integration in (10), we cannot algebraically express z in terms of x and c, we may proceed thus. From , ^_, = z, we have ax' ^zdx and finally, dx' zdz /¥)' _ [ dz f zdz '-Imlwr-m '■^'' the right-hand member indicating the performance of w — 1 successive integrations, each of which introduces an arbitrary constant. If between this equation and (10) we, after integra- tion, eliminate z, we shall obtain a final relation between y, x, and n arbitrary constants, which will be the integral sought. MakiniT nK = z, we have az ^ = V(l H-^^)? whence '^ dx" dx ^ x = c^asj[l + z') {a). 214 EQUATIONS OF AN ORDEE HIGHER [CII. X. According to the first of the above methods, we should now solve this with respect to s, and thus obtaining ^= l\(^-^ yr-j d^ A/ n ., / li > find hence ^=/V{(v)-4'^"'+''^+'^ • (*)' in which it only remains to eiFect the integrations. According to the second method, we should proceed thus. Since az dz , V(l+s) = --^ ---^^og{z-\-^/{l-\-z)]+c, CLZ dz whence multiplying the second member by -jty^^^x ^^^ ^-^^ and again integrating, 2, = «J!_|V(l+^^)log{. + V(l+^01 + |^ + ac ^/{l+z^)+c" (c). The complete primitive now results from the elimination of 2 between (a) and (c). Case III. "When the given equation merely connects two differential coefficients whose orders differ by 2. Reducing the equation to the form dy _ ^fd^'-^y dx I'fm <■* Let ^2 = z, then dz J., . This form has been considered under Case i. AKT. 3.] THAN THE FIRST, CONTINUED. 21 It gives =/ ^ , r" If from this equation z can be determined as a fimction of ir, (7, and C\ — suppose z = cf) {x, C, C), — then ^J, = {x,C,C'), the integration of which by Case I. will lead to tlie required integral. If z cannot be thus determined, we must proceed as under the same circumstances in Case ii. Ex. Gn'en«'J = ^. Proceeding as above, the final integral will be found to be Homogeneous Equations. 3. There exist certain classes of homogeneous equations wdiich admit of having their order depressed by unity. Class I. Equations which, on supposing x and 7/ to be cacli 7 J2 of the degree 1, -<- of the degree 0, -y^^ of the degree — 1, &c., become homogeneous in the ordinary sense. Adopting the notion and the language of infinitesimals, the earlier analysts described the above class of equations some- what more simply as homogeneous with respect to tiie primitive variables and their differentials, i. e. with respect to X, y, dx, dij, d'y, &c. All equations of the above class admit of having their order depressed by unity. 216 HOMOGENEOUS EQUATIONS. [CH. X. I For if we assume x = 6^, y = e^z, we shall iind by the usual f '' method for the change of variables, di/ dz . Tx = de-'' (^^)' d£'~^ KdO'"^ do) •• ^^^^' and so on. Here y is presented as of the first degree "with respect to e^ which takes the place of x, while y- is of the degree 0, and -.-{ of the degree — 1, with respect to e^. And the law of continuation is obvious. Hence, from the supposed constitution of the given equation, it follows that on substitu- tion of these values the resulting equation will be homogeneous with respect to e^, which will therefore divide out and leave , . , dz d^Z p rni an equation nivoiving only z, ,^ , -^ , &c. inat equation will therefore have its order depressed by unity on assuming J =;,.... (Art. 1.) Let us examine the general form of the result for equations of the second order. Eepresenting the given equations under the form ^(-^'i'2)- (-). we have, on substitution, and from this equation, from what has been above said, e^ will disappear on division by some power of that quantity, e.g. e"^. But the effect of simply removing a factor is the same as that of simply replacing such factor by unity. Xow to replace e"^ by ART. 3.] HOMOGENEOUS EQUATIONS. 217 unity is the same as to replace e^ by unity, and if wc do tliis 7 72 simply, i.e. without changing -^ and -^ , (17) will become ^(^'^'S+^'S+S)=« (^«)- . . ,^ dz , d'z da du , Assuming then -jn = u, whence -y^ = — - = ^^ — - , we have au do du dz Fix, z, u + z, u ^^-\-ii) = 0 (19), an equation of the first order, which by integration gives u = cj>{z,c) (20). dz Then since u = -,^ , we have du ^ = /^^^' C^^)' in which, after effecting the integration, it is only necessary to write ^ = log.T, z = ^ (22). The solution of the proposed equation is therefore involved in (20), (21), (22). Ex. Given nx^ ~^^=(y-x^]. dx- V ^^-^J Substituting as above x = e^, y = e^z, we find, as the trans- formed equation, ^d'z . dz\ /dz-^' *M TtF. + dO' ddJ \dd dz whence, making —y^ = u^ we have ^{u-J^-\-u) = ic' («), 218 HOMOGENEOUS EQUATIONS. [CH. X. which resolves itself into the two equations, ^(^1 + 1) = ^^ ^ = ^- The former gives on integration z Kow u='j^, whence do de=-^ therefore ^ = - log (^le"" + C) + C, y tion, 2' = "^^"^^!^ (i)> and now replacing 6 bj log ir, and s by - , we have on reduc- A and 5 being arbitrary constants. This is the complete primitive. The remaining equation w = 0, or ^tTi = 0, gives z = c, or y = ex, and this is the singular solution. The equation (a) might have been directly deduced from the given equation bv the general theorem (19), which indi- cates that for such deduction it is only necessary to change chi T d^y , du xiol,y to z, ~ to u + z, and -^, to u ^ -Vu, Class II. Equations which on regarding x as of the first degree, y as of the ?i*^ degree, -^ of the (?i — 1)^^ degree, -y^ of the {n — 2)^^ degree, &c., are homogeneous. To effect the proposed reduction assume a? = e^, y = e^^z. The transformed equation will be free from 6, and, on assum- ART. 3.] HOMOGENEOUS EQUATIONS. 219 dz . ing -^ = w, will degenerate into an equation of a degree lower by unity between u and z. It is easy to establish that, if the given differential equa- tion be <-^'J'S)=« (-)' the reduced equation for determining u will be CtU F[l, z, u + nz, u-^ + {2n-l) u + 7i {ri- 1) z] = 0... (24). Suppose that by the solution of this we find u = {z,c) (25), then since ^ ~ ^7^ » ^'^ have e ■ ^^ /^-' ^-)' II in which it only remains to substitute log x for 6, and j^, for z. Ex. Given c.'g = (..' + 2^2/) J -4/. This equation proves homogeneous on assuming x to be of the degree 1, 3/ of the degree 2, -,- of the degree 1, and -y^^ of the degree 0. Changing then, according to the formula (2-4), x into 1, y di/ . , d^y . dii , into z. -y- into u-{-2z, and -r^., into ii -, — \- ?>u-{- 2z, we have ax dx' dz du Uj^-\-^u + 1z={\-\- 2z) {u + 2z) - 4^' ( the only variable. Then Assume, then, removing all restriction, Subtracting this from (34) dV-dJJ^^i^j^x'^^^dx (35). 224 EXACT DIFFERENTIAL EQUATIONS. [CH. X. We remark that the highest differential coefficient -^^ has now disappeared. We observe too that the next, viz. ~ is in- volved only in the first degreet This is a consequence of the fact that the proposed differential equation was really exact. For the first member of (35) being the difference of two exact differentials, and therefore itself exact, the second member is so, and its highest differential coefficient is therefore of the first degree. The integration of an exact differential involving -r'o has, in fact, been reduced to that of an exact differential dx' involving only -^ as its highest differential coefficient. And a similar reduction may be effected whatever may be the order of the highest differential coefficient. The integration of (35) gives V-U = xy, whence A first integral of the given equation is, therefore, ^ii-^.^iiy+^^- (-)• The general rule for the integration of an exact differential dU, involving x, y, ~- , ... ^-f , is then as follows. Integrate the term which involves -—{ in the first degree, as if ^^l{ were the only variable^ and ~^, dx its differential. Representing the result hy U^, and removing the restriction, dU—dU^ will he an exact differential involving only x, y, —- , ... -ri^ • Bepeat ax ax the process as often as necessary. Then U will he expressed hi/ the sum of its successively determined portions tf^, U„, K &c. ART. 5.] EXACT DIFFEPwENTIAL EQUATIONS. 22o For tlie solution of an exact differential equation, it is there- fore only needful to equate to c the integral of the correspond- ing exact differential as found by the above process. The failure of that process, through the occurrence of a form in which the highest differential coefficient is not of the first degree, indicates that the proposed function or equa- tion is not ' exact.' 5. There is another mode of proceeding of which it is pro- per that a brief account should be given. Eepresentmg ^^, ;^, ... ^. , by ?/,,?/,, ...y„, it is easily shewn by the Calculus of Variations, that if Vdx be an exact differential, V being a function of ic, y,y^^.. .yr, , then identically dy \dx) dy^ \dx) dy^ ' ' ' \dxj dy^ ^'^ where ( — J indicates that we differentiate with respect to x regarding y^y^-t-^yn ^s functions of x. This condition was discovered by Euler. The researches of Sarrus and De Morgan, not based upon the employmsnt of the Calculus of Variations, have shewn, 1st, that the above condition is not only necessary but suffici- ent. 2ndly, that it constitutes the last of a series of theorems which enable us, when the above condition is satisfied, to reduce Vdx to an exact differential in form^ i. e. to express it in the form dU , dU . dU ' dU . ., _^,+ ^^3,+ _,/^^... + -^^^„_^ (^«)> where x, y, y^, ...3/„_i are regarded as independent. The inte- gration of Vdx = 0 in the form U=c is thus reduced to the integration of an exact differential of a function of n -f- 1 inde- pendent variables, — a subject to be discussed in Chapter xil. [Camhridge Transactions^ Vol. IX.) The condition (37) is singly equivalent to the system of conditions implied in the process of Sarrus. The proof of this equivalence a posteriori would, as Bertrand has observed, be complicated. (Liouville, Tom. xiv.) B. D. E. 15 226 MISCELLANEOUS METHODS AND EXAMPLES. [CH. X. The solution of the difFerential equations of orders higher tlian the first is sometimes effected by means of an integrating factor ^i, to discover which we might substitute fx Ffor Fin (37), and endeavour to solve the resulting partial differential equation. Even here, however, the process of Sarrus would be preferable. Miscellaneous Methods and Examples. 6. Many forms of equations, besides those above noted, can be integrated by special methods, e. g. by transformations, variation of parameters, redaction to exact differentials, &c. Equations of the classes already considered can also sometimes be integrated by processes more convenient than those above explained. ,^ ^. d^i/ , Ex. I. Given -^ = ax -\- by. Let ax + l)y = t. We find as the result, — ^^ = ht d^ linear equation with constant coefficients. Ex. 2. Given (1 - ^') g - ,. J + q^y = 0. Changing the independent variable by assuming sin~^a; = t, d^V we find -yy + fy = 0, whence the final solution is y = c^ cos {q sin~^a;) + c^ sin {^ shr^a:) (39). 72 7 So too the equation (1 + ax^) ^ +'^^T" ± O = ^> is re- d^7i ducible to the form — ; ± q'y = 0, by the assumption /: dx _ V(l + ctx') Equations involving the arc s, whether explicitly or im- plicitly, may be freed from it by differentiation or by change^ of independent variable. ART. 7.] MISCELLANEOUS METHODS AND EXAMPLES. 227 Ex. 3. Given s = ax-\- hy. Differentiating, we have a / U + ( "r. ) [ = « + ^ -r ; , ^ dy al>±^/{a'+}r-\) thcrctore --r = r — r' ~ > dx I — 0" y= ~ i_i/ ^ + ^- d^'x Ex. 4. Given ^t = «. ds' Assuming x as independent variable, we liave d'x _dx d^dx_ fdsY d fds^ ds^ ds dx ds \dxj dx \dxj ds\' d-'s _ dx) dx^ ds We miglit here put for -^^ its value \![\ +2'*')^ ^^^ ^^ ^^^"^ a differential equation for determining p. Direct integTation, however, gives F = 2ax + c. Whence we find d!l_( 1 _iM dx \2ax + c y= \ { \ I) dx-^ c ', which indicates a cycloid. 7. M. Liouville has shewn how to integrate the general e([uation -fi+f{x) ^ + F{y) (~t ] = ^> (Journal de Matlicma- tiques, 1st Series, Tom. vir. p. 134). Suppressing the last term, the resulting equation 15—2 228 MISCELLANEOUS METHODS AND EXAMPLES. [CH. X. has for a first integral -,- = Q^-ff(x)dx^ -^^^ assume this to be ^ ax a first integral of the given equation regarding C as an un- known function of?/, then ~ G'cbj\cU -^^^dx' Thus, the given equation becomes ]M^^^=' ^''^' whence C = Ae-J^'y^''->. ' Therefore '^ = Ae'^^''^^ ''' x e"-^^*^) '^ ; ax therefore ^ e^^^y^'y dy = a\ e'^^^''^'^ dx-\-B (41), the complete primiti^'e sought. 8. Jacobi has established that when one of the first inte- grals of a differential equation of the form ~-j\_=f{x-> y) is known, the com]:)lete primitive may be found. The following- demonstration of this proposition is due to Liouville, [Journal de MatJmnatiqiies, 1st Series, Tom. XIV. p, 225). n nrsi miegrai oe ■ ' atins:, ^yQ have Let the given first integral be -/- = (f> {^, y-> c). DifFerenti- d'^y _ dxj) d(f) dy _ d6 , dcj) dx"^ dx dy dx dx ^ dy ' (^ standing for ^ {x. y, c). Hence, comparing with the given equation, d<^ , dd> ., . ART. 9.] SINGULAR INTEGRALS. 229 and difFerentiatlng with respect to c, dxdc dc dy dydc Now this is precisely the condition which must be satisfied in order that the expression -—- {dy — ^dx) may be an exact dlfferentiah Hence, the first integral expressed in the form dy— (j)dx = 0, is made an exact differential by means of the factor -~ . The complete primitive therefore is / fjdy-dx) = c (42), Some equations of great difficulty connected with the theory of the elliptic functions are reduced to the above case in the memoir referred to. Singular Integrals, 9. Equations of the higher orders, like those of the first order, sometimes admit of sijigular integrals, i. e. of integrals not derivable from the ordinary ones without making one or more of their constants variable. We shall term such integrals singular solutions when they connect only the primitive variables, but singular integrals when they present themselves in the form of differential equations inferior in order to the equation given. And as the entire theory is involved in the theory of singular first integrals, we shall speak chiefly of these, but with less detail than in the corresponding inquiries of Chap. Yiii. Prop. Given a first integral with arbitrary constant of a differential equation of the ?i^'' order, required the correspond- ing singular integral. Let the given equation be ^(^,^,y.,y....yn)=0 (43), 230 SIXGULAR INTEGRALS. [^^'H. X. 7 72 where y^ stands for ~- , y^ for -y^ , &c. Suppose the integral given to be expressed in the form yn-x^f{^, y. y, "-yn-,, c) (44), c being an arbitrary constant. Differentiating as if c were an unknown function of x, _(lf_^df df_ Jf_ df dc^ y^"' dx^ dy^''^ dyj^''"'^ dy^^'^''-'^ dc dx' Kow this reduces to the same form, i.e. gives the same expression for ?/„ in terms oi x, y ... y„_^, c, as it would do if c were constant, provided that we have -7-=0; and therefore, tliis condition satisfied, the elimination of c will still lead to the given differential equation (43). An integral of the given equation will therefore be found by attributing to c in the complete first integral (44), sucli S Wlii SctLlteiJ' LiiC CUllUlLiUIl press it value as will satis-fj the condition ^ = 0, or, as we may ex- %'=« (-)■ And unless the value of c thus found is constant, the in- tegral will be singular. The above process amounts to eliminating c between (44) and (45), so that we have the following rule. Given a first integral of a differential equation of the n^^' order, to deduce the corresponding singular integral, we must eliminate c hetween the first integral in question and the equation -- "— _ 0^ icJiere y^^ is the value of ^ „:^ expressed in terms of x, y ... .^^ , iScc. l>y means of the given first integral. If the proposed first integral is rational and integral in form, ART. 9.] SINGULAR INTEGRALS. 231 then representing it hy ^ = 0, it suffices to eliminate c between the equations, '^='>'S=o • (■•«)• It is unnecessary to dwell on the particular cases of exception after what has been said on tliis subject in Chap. Viii. Ex. 1. The differential equation y - ^y.^ \,: y^- (]j,- ^yj'' - y' = ^^ has for a first integral required the corresponding singular integral. Differentiating the first integral with respect to h, we find whence h — — jy — 2. , and this value substituted in the given integral, leads to or, on reduction, 16 (1 + x") 2j - d>x^ij^ - l^xy^ + x* - l(jy^ = 0. In connexion with this subject, Lagrange has established the following propositions : 1st. Either of the first two integrals of a differential equation of the second order leads to the same singular integral of that equation. 2nd. The complete primitive of a singular integral of a differential equation of the second order will itself be a sin- gular solution of that equation, but a singular solution of a singular integral will in general not be a solution at all of that equation. 232 SINGULAR INTEGRALS. [CH. X. The proof of these propositions will afford an exercise for the student. [See Lagrange's Legons sur h Calcul des Fonctions, Lecon 14°"" of the edition of 1806, or Lecon 15°"" of the edition of 1808. A note by Poisson on page 239 of the edition of 1808 should be consulted ; it relates to the second of the above two propositions. See also Lacroix, Tome ii. pp. 382 and 390.] 10. We proceed to inquire how singular integrals may be determined from the differential equation. Expressing as before the first integral involving an arbitrary constant in the form yn-,=f[^,y,yr'-'yn..,c) (^7), we have as the derived equation ^^^j€(-^c:.y^)| (48), the brackets in the second member indicating that in effecting the differentiation y, ?/j,...y„ g, are to be regarded as func- tions of X. The differential equation of the n^^^ order is found from (48) by substituting therein, after the differentiation, for c its value in terms of x, 3/, 3/^, ... y„_i, given by (47). The result assumes the form yn = i>{x, y,yx'"yn.,) (49). Hence, we liave f^^^ in (49) = -^ in (48) x -^ in (47), ^3/»_i ' ' dc df/„_^ or, representing f{x, y, y^.-.y^.^, c), by/, dy, . _(dj\,df dij^_^ \dxdcj ' dc' Hence, ^.=(i^°="%') ("^' provided that the first member be obtained from the differ- ential equation, and the second member from one of its ART. 10.] SINGULAR INTEGRALS. 233 first integrals involving c as arbitrary constant. It is to be borne in mind that in effecting the differentiation with respect to X in the second member, we must regard y^ t/^, ... y^_^ as functions of x, Xow reasoning as in Chap. viii. since a singular solution makes -^^ = 0, it makes its logarithm, and in general the differential of its logarithm, infinite. Thus we arrive at the following conclusion. k. singular integral of a differential equation of the rt^^ order (ifii will in general satisfy the condition -~^ = qc , and a relation which satisfies both this condition and tiie differential equation will be a singular integral. Ex. 2. Applying this method to the equation, x" y - ^Ux + ~^ y^- ^yx-^y^^-yi=^^ we find, on differentiating with respect to y^ and y^ only, -1^+2 [y^ - xy^)] fhj^ + 1^ + 2x (y, - xy^ - 2^ j dy^ = 0, whence dy„ 2 (?/j — .-TT/J + X '^•^^ ^^- + 2x{y^-xy,)-2y. Equating the denominator of this expression to 0, we find _ a;" + 4.T?/j ^'~^{x' + 'l)' and substituting this value in the given differential equation, clearing of fractions, and dividing hj x~+l, which will present itself as a common factor, Ux^y + 16?/ - Sx^'y^ - Uxy^^ - Uy^" + a:' = 0, a singular integral. The equation given and the result agree with those of Ex. 1. 234 EXERCISES. [CH. X. 4: EXERCISES. EXERCISES, 1. d'y , . dx^ 2. d'^y -a dx' {2ax-xy 3. dSf _ 1 dx' ~ ^{ay) • 4. dx' X dx 5. dh/ 2?/ dx' ~ x'' The two following are reducible to Clairaut's form. dx dx-' -^ \dx\ \dxj "^ dx' dx-^ {\dxj dx\ 10. Describe tlie different kinds of homogeneity in differ- ential equations, and explain their connexion. The two following homogeneous equations are intended to be solved bj the method developed in Art. 3. 1^- ""d^^^-'^dx ^-'- ^ dJ ^ dx ^ \dx) ^^ ' CII. X.] EXERCISES. 235 13. Shew tliat tlie linear equation . '^ + P ^ + Cy =-0, belongs to one of the homogeneous classes, and is reducible to an equation of the first order by assuming y = e-^"'^"'. 14. bolve the hnear equation ^.^ ■\- F-j--\- -j-- y = 0. 15. Mainardi has remarked (Tortolini, Vol. i. p. 7G), that LiouvIUe's equation Art. 7, becomes integrable if multiplied by the factor [-j-] . Applying this method, deduce the com- plete primitive. 16. Liouville's equation may also be solved by suppressing the second term and regarding the arbitrary constant in the first integral of the result as an unknown function of x. 17. Shew that the equation y^ + -P -7^ + . /jy U — (x •" (t) 7} Perpendicular from (a, V) on tangent = ^^ n~4- ^u^ • Perpendicular from (a, V) on normal = /TT^syr^ • ART. 2.] GEOMETRICAL APPLICATIONS. 239 Jtadius 01 curvature = + ^ ^^-^ . 9. Co-ordinates (a, /3) of centre of curvature To tliese may be added the well-known formulae for the dif- ferentials of arcs, areas, &c. It is evident from the above forms that problems which relate only to direction or tangency, give rise to differential equations of the first order — problems which involve the con- ception of curvature to equations of the second order. When tlie conditions of a geometrical problem have been expressed by a differential equation, and that equation has been solved, it will still be necessary to determine the species of the solution — general, particular, or singular, as also its geometrical significance. 2. The class of problems which first presents itself, is that in which it is required to determine a family of curves by the condition that some one of the elements whose expressions are given above shall be constant. Ex. 1. Eequired to determine the curves whose subnormal is constant. Here y -~ =a, and integrating, f 'j-=rax-\-c, y = {'lax + 2cf. The property is seen to belong to the parabola whose para- meter is double of the constant distance in question, and whose axis coincides with the axis of x, while the position of the vertex on that axis is arbitrary. Ex. 2. Eequired a curve in which the perpendicular from the origin upon the tangent is constant and equal to a. Here we have y-xp = a{\ -}-/)S 240 GEOMETRICAL APPLICATIONS. [CH. XL an equation of Clairaut's form, of which the complete primi- tive is and the singular solution The former denotes a family of straight lines whose distance from the origin is equal to a, the latter a circle whose centre is at the origin, and whose radius is equal to a. And here, as was noted generally by Lagrange, the singular solution seems to be, in relation to geometry, the more important of the two. 3. A more general class of problems is that in which it is required to determine the curves in which some one of the foregoing elements. Art. 1, is equal to a given function of the abscissa x. Ex. 1. Eequired the class of curves in which the subtan- gent is equal tof(x). Here we have whence 2/=/(^)J; Ji/ dx Thus if the proposed function were x^, we should have 1 as the equation required. Ex. 2. Eequired the family of curves in which the radius of curvature is equal to / {x) . Here we have d^ dx' _ 1 1 + {MV /W ART. 4.] GEOMETRICAL ArPLICATIOXS. 241 whence, multiplying by dx and integrating, dy dx __ f dx dx X representing the integral of , . Hence we find by alge- braic solution dy X+G dx~{i-{x+cy]'' 'k (X + C) dx ^ in Avhich it only remains to substitute for X its value, and effect the remaining integration. If/ {x) is constant and equal to a, we find X+0=-+(7 = ^±^, a a _ r (x + aC) dx ^ y " J J\a'-ix-\-aGY\ ^ ^^' sJ[cc^-{x-\-aCY =.-[a'-{x + aCyf+C„ whence {y — CJ^+ {x + a Cy= d\ and tliis represents a circle whose centre is arbitrary in posi- tion, and whose radius is a. A yet more general class of problems is that in which it is required that one of the elements expressed in Art. 1 should be expressed by a given function of x and ?/. An example of this class is given in Chap. vir. Art. 10. 4. We proceed in the next place to consider certain pro- blems in which more than one of the elements expressed iu Art. 1, are involved. B. D. E. , 16 242 GEO:\IETRICAL APPLICATIONS. [CH. XI. Ex. 1. To determine the curves in wliicli tlie radius of curvature is equal to the normal. If the radius of curvature have the same direction as the normal we shall have 1 + r^^iT 1+ £ (1). whence dx' y'MS-^-^ (^)- The first side multiplied by dx is an exact differential and gives dy whence again integrating y' + x' = 2cx-\-c (3), the equation of a circle whose centre is on the axis of x. If the direction of the radius of curvatm-e be opposite to that of the normal, it will be necessary to change the sign of the first member of (1). Instead of (2) we shall have 4-J-(i)"-» ■■-«. and this equation not containing x, we may depress it to the first order by assuming -r —p- The transformed equation is , pdp dy whence, ■ , , ., = — 2/ = c(l+/)l AllT. 4.] GEOMETEICAL APPLICATIONS. 243 Substituting for i) its value -j- , we find on algebraic solu- tion d^= "^y whence, x = c' + c\og [1/ + {7/^ — cy-}... (5). This equation, reduced to the exponential form a ■ "-' y = 2^^"+^ ")' (6). is seen to represent a catenarj. The solution therefore indicates a circle when the direction of the radius of curvature and of the normal are the same, but a catenary when they are opposed. The latter curve has, liowever, many properties analogous to those of the circle. (Lacroix, Tom. ii. p. 459.) Ex. 2. To find a curve in which the area, as expressed by the formula fyclx, is in a constant ratio to the correspond- ing arc. We have y = C {I +7->^)% which, agreeing in form with tlie last differential equation of the preceding problem, shews that (5) represents the curve required, and connects together the properties noticed in the last two examples. Ex. 3. Required the class of curves in which the length of the normal is a given function of the distance of its foot from the origin. The differential equation is y{l+ff=f{x + yp) (1), and it belongs to the remarkable class discussed in Chap. vil. Art. 9, where the complete primitive is given, viz. y-+{x-af==[f{a)Y (2). This represents a circle whose centre is situated on the axis of a; at a distance a from the origin, and whose radius is equal to IG— 2 2i4 GEOMETRICAL APPLICATIONS. [CH. XL /(«). It is evident that this circle satisfies the geometrical conditions of the problem. But there is also a singular solution, found hy eliminating the constant a between (2) and the equation derived from (2) by differentiation with respect to a, viz. x-a+f{a)f{a) = 0 (3). For instance, if f{a) = ivd' we have to eliminate a between the equations ?/^ + (a? — CLf- = ??«, 2 (x - a) 4- n = 0, from which we find the equation of a parabola. While in this example the com- plete primitive represents circles only, the singular solution represents an infinite variety of distinct curves, each originat- ing in a distinct form of the function /(a). Other illustrations of this remark will be met with. The above problem was first discussed by Leibnitz, who did not, however, regard its solution as dependent upon that of a differential equation, but, establishing by independent con- siderations the equation (2), which constitutes in the above mode of treatment the complete primitive of a differential equation, arrived at a result equivalent to its singular solu- tion by that kind of reasoning which is employed in the geo- metrical theory of envelopes. Indeed it w^as in the discussion of this problem that the foundations of that theory were laid (Lagrange, Calcul des Fonctions, p. 268). 5. A certain historic interest belongs also to the two fol- lowing problems, famous in the earlier days of the Calculus, viz. the problem of ' Trajectories' and the problem of 'Curves of pursuit.' These we shall consider next. They will serve to illustrate in some degree the modes of consideration by wliich the differential equations of a problem are formed when a mere table of analytical expressions suffices no longer. AIJT. 5.] TRAJECTORIES. 245 Trajectories, Supposino^ a system of curves to be described, the different members differing only through the different values given to an arbitrary constant in their common equation — a curve which intersects them all at a constant angle is called a trajectory, and when the angle is right, an orthogonal trajectory. To determine the ortho_2:onal trajectory of a system of curves represented by the equation [x,y,c)=0 (1). Eepresenting for brevity (/> {x, y, c) by (/>, we have on differ- entiating -J- ax + -r ay = 0. ax dy "^ Hence, for the intersected, curves, dy _ d^ dcf) dx dx ' dy ' !N"ow representing this value by m^ and the corresponding value of -~~ for the trajectory by m\ we have, by the condition of perpendicularity, m = — . Hence for the trajectory dy d(j) dcf) dx dy ' dx^ d(f) , d(b 7 ^ , V T/^'^-Zi'^^^'' ' ("^)' which must be true for all values of c. Hence the differential equation of the orthogonal trajectory icill he found hy elimin- ating c hetioeen (1) and (2). Were the equation of the system of intersected curves pre- sented in the form {x, y, «, h) = 0, a and h being connected by a condition f{a,h)=0, 246 TRAJECTORIES. ' [CH. XT. we slionlcl have to eliminate a and h between tlie above two equations, and the equation d4> (x, ?/, a, h) ^^^ _ d4> {x, ?/, a, h) ^ ^^ ^^U dx "\Vc shall exemplify both forms of the problem. Ex. 1. Required the orthogonal trajectory of tlie system of curves represented by the equation y = cx'\ Here <^=y — ex"", whence by (2) dx + ncx''~^ dy = 0. Eliminating c, xdx + nydy — 0 ; therefore x^ + nif = c, the equation required. We see that the trajectory will be an ellipse for all positive values of n except ;i = l, — an ellipse, therefore, when the intersected curves are a system of common parabolas. The trajectory is a circle if w = 1, the mtersected system then being one of straight lines passing -through the origin. The trajectory is an hyperbola if 7i is negative. Ex. 2. Required the orthogonal trajectory of a system of confocal ellipses. The general equation of such a system is •> 1^ 7 2 — -*■* a 0 a and h being connected by the condition 2 7 2 7 2 a — b — h , Avhere li is the semi-distance of the foci, and does not vary from curve to curve. Hence we have to eliminate a and h from the above equations, and the equation f ,- dx - - 2 dy = 0 ; ART. C] TRAJECTORIES. 247 the result is the solution of which may be cleduced from that of Ex. 3, Chap. yii. Art. 10, by assuming therein A = \, B = h^. We find and this may be reduced to the form a^ and h^ being connected by the condition Thus the trajectory is an hyperbola confocal with the given system of ellipses. 6. When the trajectory is oblique, then 6 being the angle which it makes with each curve of the system, and m and m having the same significations as before, m + tan 6 m 1-m tan 6 ' or, substitutins: for m its former value — r~^ ~r- i f^nd for m ^ ax ay its value ^- as referred to the trajectory, we have on reduction ^ #tan^-4^ dx 716 d6 ^^' ay dx an equation from which it only remains to eliminate c by means of the given equation in order to obtain the differential equation of the trajectory. Ex. Kequired the general equation of the trajectories of the system of straight lines y — ax. 248 TRAJECTORIES. [CH. XT. Here cj) = y — ax, whence by (3) di/ _ tan 6 + a dx 1 — a tan Q _ X tan 0 ■\- y X — y tan 6 ' or {y-Vx tan 6) dx + {y tan 6 — x) dy = 0, a homogeneous equation, an integrating factor of which being T, , we have 03^+2/ ydx — xdi/ , ^ a?f?ic + ?/cZ?/ ^— ij f^ + tan ^ — r, — V^ = 0, x' + y' x' + y^ ' whence integrating tan-^ - + tan (9 log {x' + 2/')^ = c. If we change the co-ordinates by assuming x =r cos , 2/ = r sin , we get the equation of a logarithmic spiral. The following example, which is taken from a Memoir by Mainardi (Tortolini's Annali di Scienze Matematiche e Fisiclie, Tom. T. 251), is chiefly interesting from the mode in which the integration is effected. Required the oblique trajectory of a system of confocal ellipses. Eepresenting the tangent of the angle of intersection by w, we have to eliminate a and h between the equations y X 0 a i5 + ™a^ ART. C] TRAJECTORIES. 249 The result may be expressed in the form [nx + 2/ + [ny — x) p] [x - ny + {nx + y) p] - h^ {n —p)) (I + np^ . To integrate this equation let us assume x — ny-\- [nx + y) p = M {I + np) , M{nx + y + {ny — x)p]=h^ [n—p). As these on multiplication reproduce the given equation the assumption is legitimate. Eliminating p from the last two equations, and dividing by 1 + n^, we have [x'+if + h')2I=x{M'' + li') (a). Differentiating this equation and eliminating y and|9 from the result by the aid of any two of the last three equations (it is evident that two only are independent), we obtain a differential equation between M and x, which is capable of expression in the form nd [xM) x _ [K\xM) - {xMf^^ ' Mf_M\^ X\ X J [For (a) may be written thus : ]\hf={x-M)[h'-xM) (c); differentiating we have therefore 2, f Jf + ^-^^)^--^^ if ^ dx M dx 250 TRAJECTORIES. [CH. XI. therefore 2y £m^ ^^ -^- = ]^ _ ^ii/_ (^ _ J/) (mjt X ^) ; therefore 2?/ — il/+ -^—^ ^ -— = h'- 2x21 +M\ But ^= ^^+7"^ , therefore 2% (J/- iz?) + 2?zJ/y + ?^ (a^ - il/) \h' ~ 2xM + il/^ - jj. {h' - 2P) '^~ I ; therefore y (i)p _ 7,^) + 2,1 {x - 21) {h' - x2r) + 'H (A^ - 2P) ~ = n{x- 21) j/r - 2x21+ 2P -j^iji'- 2P) ^l ; tlierefore {2P - 7r) (^^ - «^ ^^) - n (x - 21) {2P - h') = n(x-2I)l^iM^-P)^; therefore '' (^- -^^) [^^+ ^^) + 2/ (x^ - 2lj = 0; therefore .i (x - 21) '^-^-^ + yx' ?- ^^= 0. c/a; "^ dx X AKT. 7.] TEAJECTOEIES. 251 Hence by t]ie aid of (c) we obtain (5).] Hence, by integration '" *"^" \/{&r ') + ^°s ^, 1^. = c, in which it is only necessary to substitute for 21 its value in terms of x and y deduced from (a). Curves of Pursuit. 7. The term curve of pursuit is given to the path which a point describes when moving with uniform velocity toward another point which moves with uniform velocity in a given curve. Let X, y be the co-ordinates of the pursuing point, cc', y the simultaneous co-ordinates of the point pursued. Also let the equation of the given path of the latter be /(^',y)=o (J). Now the point pursued being always in the tangent to the path of the point which pursues, its co-ordinates must satisfy the equation of that tangent. Hence, y -y^dx^"" -^) W- Lastly, the velocities of the two points being uniform, the corresponding elementary arcs will be in the constant ratio of the velocities with wliich they are described. Hence, if the velocity of the pursuing point be to that of the point pursued as 11 : 1, we have n ^{dx" + chf) = >^{dx' 4- fZ/), or, taking x as independent variable, ■V{®)"-(S}Vl-0') <«• 252 CURVES OF PURSUIT. [CII. XI. the sign to be given to each radical being positive or negative, according as the motion tends to increase or to diminish the corresponding arc. From (4) and (5), when the form of the fmiction / (a;', y) is determined, x and y maj be found in terms of a?, ?/, and -,-^ , and these values enable us to reduce (6) to an equation be- tween ic, ?/, -^ , -j^ . It only remains to solve this diifer- ential equation of the second order. If the signs of the radicals are both changed, the motion in each curve is simply reversed, and the curve of pursuit becomes a curve of fliglit. But the differential equation remaining unchanged, the forms of the curves are unchanged, and only their relation inverted. Ex. A particle which sets off from a point in the axis of x^ situated at a distance a from the origin, and moves uniformly in a vertical direction parallel to the axis of ?/, is pursued by a particle which sets off at the same moment from the origin and travels with a velocity which is to that of the former as n \ \. Required the path of the latter. The equation of the path of the first particle being x = a, (5) becomes whence Thus we have dx ^ dx~^ ^ dx- and the differential equation, both radicals being positive, is ART. 8.] CCRVES OF PURSUIT. Hence, v/h©] •"-"• Multiplying by die and integrating I4m"-)---J therefore -- = - {c (a - r?^) ---[a-x) Hence, if n be not equal to 1, But if ?i be equal to 1, we have dy _\ [x — a c ] dx 2 ( c ic — rtj ' whence (x — aY c , , . , 8. The class of problems which we shall next consider is introduced chiefly on account of the instructive light which it throws upon the singular solutions of difterential equations of the second order. Inverse Problems in Geometry and Optics. The problems we are about to discuss are the followiuo: : 1st, To determine the involute of a plane curve. 2ndly, To determine the form of the reflecting curve which will produce a given caustic^ the incident rays being supposed parallel. 254 PEOBLEMS [CH. XI. In botli tliese problems we shall have occasion in a parti- cular part of the process to solve a differential equation of the first order of the form y-x

{p)-{p)f-'4> ip) (7), in which (/> and/ are functional symbols of given interpretation, and /'"' is a functional symbol whose interpretation is inverse to that of the symbol/'. Thus, i^fix) = sin^, then f {x)= cos X, f'~^ {x) = cosT^x, It will somewhat less interrupt the theoretical obser- vations for the sake of which the above problems are chiefly valuable, if we solve the equation (7) under its general form first. Eeferring to Chap. vil. Art. 7, we see that (7) will become linear if we transform it so as to make either of the primitive variables the dependent variable, and either p or any function of p the independent variable. Let us then assume and transform the difi'erential equation so as to make x and v the new variables. Substituting v for (p {p) in (7), we have y-xv=fr{v)-vf-^[v) (8). Differentiating, and regarding v as independent variable, But dy dx ,_. , , dx ART. 8.] IN GEOMETIIY AND OPTICS. 255 Hence, or, ^ _|. ^ ^ /'"' (^) Hence, if for brevity we write we have cc = 6 - "/^ '^') { (7 + Je'/' '^) ^/r' (f) / '-^ [v) dv} = €-'/' >; [C+e^ '^^ /'-' {v) - 1 6^ :^') df'-' {v)}, whence ^-/'"'('•)=e-*'^'((?-/^*''-'<^/'-'Wl (10), between which and (8), v must be eliminated. If in those equations we make /'"^ {v) = t, they assume the somewhat more convenient form, 2/-^/'w=/w-r(o> and these may yet further be reduced to the form a,^t^?L^m= J lu). From these equations it only remains to eliminate f, tlic forms of/ and being specified, and that of yjr given by (9) ; and this is apparently the simplest form of the solution. 256 PROBLEMS .[CH. XI. 9. We shall now proceed to the special problems under consideration. To determine the involute of a plane curve. It is evident from the equations which present themselves in the investigation of the radius of curvature, that if x, y be the co-ordinates of any point in a plane curve, and a?', y those of the corresponding point in the evolute, then 7 72 where ;:> = -^ , 2' = 7^ (Todhunter's Differential Calculus, Art. 320). Hence, if the equation of the evolute be y=/(^') (12), we shall have on substituting therein for y' and x the values above given, y^^-f^-^^ (13), a differential equation of the second order connecting x and y, and therefore true for each point of the curve whose evolute is Liiven. Of that evolute the curve in question is an involute. Hence, \iy'=f{x') be the equation of a given curve, the equation of its involute will satisfy the differential equa- tion (13). Xow suppose that nothing was known of the genesis of the above equation, and that it was required to deduce its complete primitive, and its singular solution, should such exist. Upon examination the equation (13) will prove to be of a kind analogous to that of Chap. Yii. Art. 9. If we assume x-P'^^P'^a (U), 2 l + if ■ . y+'^-i (15), ART. 9.] IN GEOMETRY AND OPTICS. 257 a and h being arbitrary constants, we shall find that each of these leads by differentiation to the same differential equation of the third order, viz. 3^2'-(l+/)^=0 * (16), where r stands for -r-^ . It follows hence, that a first intesrral dx of (13) will be found by eliminating q between (14) and (15), and connecting the arbitrary constants h and a by the relation h=f[a). Eliminating q, we find x-a + {y-h)p = 0 (17), wherein making h=f{a), we have ^-«+{y-/(«)b = 0 (18), for the first integral in question. Again, integrating, we have {x-ay+y-f{a)Y = r^ (19), in which a and r are arbitrary constants. This is the complete primitive of (13). It is manifest from its form that it repre- sents, 'not the involute of the given curve, but the circles of curvature of that involute. Indeed, that the complete primi- tive cannot represent the involute might have been affirmed a priori. The equation of the involute of a given curve cannot involve in its expression more than one arbitrary constant ; for the only element left arbitrary in the mechanical genesis of the involute is the length of a string. It remains to examine the singular solution of (13). This is most easily deduced by eliminating a between the first integral (18) and its derived equation with respect to a, viz. between the equations a^-« + {/y-/(^)h^ = 0 (20), -l-/^(«)i^ = 0 (21). From the second of these we have B. D. E. 17 258 PROBLEMS [CH. XI. Hence eliminating a from (20) x + yp=r{~j)+pff-(~) (22), whicli is the singular solution of (13), and the differential equation of the first order of the involute sought. This equation is a particular case of (7). If we express it in the form we see that it is what (7) would become on making Hence comparing with the general solution (11) we have ^(^)=JiL=iog(.^+i)^ v + V Thus the system (11) becomes "-'- fit) uTfw¥~" ^''^- The final solution is therefore expressed in the following theorem. Given the equation of a curve in tlie form y' = f {x) , that of its involute is found hy eliminating t from the system (23). 10. Parallel rays incident, in a given direction, on a reflect- ing plane curve produce after reflection a caustic whose equa- tion is given. The equation of the reflecting curve is required. ART. 10.] IN GEOMETRY AND OPTICS. 259 Let ZPbe a ray incident parallel to the axis of a? on a point P in tlie reflecting curve SPM, Fig. I, PF Q the reflected ray cutting the axis of cc in Q and touching the caustic S'P'M' in P'. Let X, y be the co-ordinates of P, x\ y those of F. Let the equation of the caustic be y' =f[x'). It is an easy consequence of the law of reflection that the angle PQX which the reflected ray makes with the axis of x is double of the angle PTX made by the tangent at P with tlie axis of x. This at once gives us the equation y — y _ 2j9 vv^here _p = y • Hence y-y'-J^zi^-^') = (^ (24). As, however, {x, y') is a point at which consecutiye re- flected rays intersect, we are permitted to difterentiate the above equation regarding x and y' as constant while x and y vary. We thus obtain, representing -^ ^7 9.1 1 — 'p ^ (1 —;p) 1 ' 7^(1-/) whence x — x= — ^ ^ , and a:' = aj+£iLlZ) (95). Substituting this value in (24), we have y ^-rz/'' 2^ -"T"' 2 whence y' = y -\-'^- (26). 17—2 260 PROBLEMS " [CH. XI. Were tlie equation of the reflecting curve given and that of the caustic required, it would only be necessary to substitute in (25) and (26) the values oijj and q in terms of x and y derived from the former, and then by eliminating x and y from the three, to deduce the relation between x and y'. Conversely, to determine the reflecting curve we must elimi- nate X and y' from (25), (26) and the equation of the caustic, viz. y =f{^')' ^'l^e result which is obtained by mere substi- tution is »-?=/!-'-V=} w. a difl'erential equation of the second order, the solution of which will determine in the fullest manner the possible rela- tions between x and y which are consistent with the conditions of the problem. Were this equation given and nothing known respecting its oria'in, we might at once infer that it is of a class analogous to those of Chap. Vii. Art. 9. For writing 3/+f = J, -+^^V^=« (28), we find that each of these leads by differentiation to the same differential equation of the third order. For the first gives 9 pr _ jjj while the second oives 3i-^^ = 0, 3 3 {\-f)vr_ 2 2^ 2q' ~^' and these lead to the same value of the differential coefficient of the third order r, viz. V this constituting the essential criterion of agreement between differential equations of the third order. ART. 10.] IX GEOMETRY AND OrilCS. 2G1 Accordingly, eliminating ri from (28) and afterwards making })=f{a) by virtue of (27), we find f{a)-y 2{a-x)' <^r 2/-/(«)=^~.(^-«) (29), wliicli is a complete first integral of (27). We see that it agrees, and necessarily so, with (24), a only taking tlie place of x and /(a) that of y. The complete integral of (29) will he found to be {y-fia)Y = Am(x-a)-\-4.m' (30), 771 being an arbitrary constant. And this is the complete primitive of (27). If we substitute x for o, wliich we may without loss of generality do, then f{a) =f[x') =9/', so that the above equation gives {y - y'Y = 4.m{x-x' + m) (31); and this is evidently the equation of a parabola whose axis is parallel to the axis of x, whose focus is upon the caustic curve, but which is in no other way limited. The complete primitive of (27) represents then a system of such parabolas. It is plain that any such system does constitute a true solu- tion of the problem, rays falling upon the interior arc of a parabola, and parallel to its axis, being accurately reflected to the focus. It remains to deduce the singular solution of (27). Differ- entiating its first integral (29) with respect to a, we have whence a=f'~'(~^^, and substituting this in (29) 262 PROBLEMS IX GEOMETRY AND OPTICS. [CH. XI. This is the differential equation of the involute. Its com- plete integral may be deduced from tlie general solution in Art. 8, by making cj) {p) = . _^ 2 , whence we have = log(^/{l + ^-•^)+ll. Hence the system (11) becomes ^^t- y-/W _ ^-/[i + V(i +/' [tfW dt fit) i+vii+/'m "•^^^^' froni which, after the integration has been effected, t must be eliminated. If, as before, we replace t by x\ and./{t) by y' and there- fore /' {t) by y, , then, since we have \^[li-f'{ty]dt = ds', where s' represents the arc of the caustic, the above system assumes the following form, , V — V C — x — s ,^,. """^^^"77*:" ^ ^' dx' dx from which, when s' is determined, x and ?/' must be elimi- nated by means of the equation of the given curve. From the above it appears that, the incident rays being parallel, the reflecting curve can always be determined when the caustic can be rectified. We see also from the nature of the connexion between the singular solutions and the ordinary primitives of differential equations, that the reflecting curve is in reality the envelope of ART. 11.] INTRINSIC EQUATION OF A CURVE. 263 a system of parabolas whose axes are parallel to tlie direc- tion of incident rays, whose foci are on the caustic, and whose parameters are subject to such a relation as makes that envelope to have contact of the second order with the curves out of whose differential elements it is formed. It is not merely an envelope, but an osculating envelope. Analogy makes it evident that when the rays instead of being parallel issue from a given point, the reflecting curve is the osculating envelope of a system of ellipses, each of which has one focus at the radiant point, and the other on the arc of the caustic, the elliptic elements being further so conditioned as to render such osculation possible. Lastly, it is plain that the problem of caustics in its direct and in its inverse form, as stated above, is in strict analogy with the direct and the inverse form of the problem of curva- ture, osculating parabolas and ellipses occupying the place and relation of osculating circles. The above examples might also be treated by a remarkable method, the consideration of which will fitly close this Chapter. Intrinsic Equation of a Curve. 11. There are certain problems, the solution of which is much facilitated by the employment of what Dr Whewell has happily termed, the intrinsic equation of a curve, viz. the equation which expresses the relation between tlie length of an arc and the anHe throuirh which it bends, the latter beins: in more precise language the angle of deviation of the tangent from the tangent at the origin. These elements are called intrinsic because they are independent of any external lines of reference, and it will be noted that they form a system dif- fering essentially from all systems of co-ordinates which begin by the defining of the position of a point, and in the applica- tion of which a curve is contemplated as a collection of points. The conceptions of lengtli and deviation upon wliich the above system is founded, might be replaced by the not less fun- damental conceptions of length and curvature, tlic equation of the curve being then expressed in terms of its radius of curva- ture at the extremity of an arc and the length of that arc. Or, 264 INTRINSIC EQUATION OF A CURVE. [CH. XI. in place of either of these systems, we might employ that which defines a curve by the relation which connects the cm-vatm-e at any point with the deviation of the tangent. Of the three elements, length, curvature, and deviation, any two indeed will together constitute an equivalent system. Euler, in a particular class of problems, employed the combination last described. Here we shall select the one first mentioned, and shall borrow our chief illustrations of its use from the memoir of Dr Whewell ( Cambridge Philosoj)Mcal Transactions^ Vol. VIII. p. 659, and Vol. ix. p. 150). Kepresenting by s the variable length of an arc the begin- ning of which is assumed as origin, and by (/> the corresponding angle of deviation, the intrinsic equation is of the form s=f{^) (35). Thus in fig. 2, /SP=5 and ATS= 0. From this equation the ordinary equation in rectangular co- ordinates may be found in the following manner. Still taking the beginning of the arc as origin, let the tangent at that point be taken as the axis of x, then will the element of the curve ds be inclined at an angle ^ to the axis x. Its projection on the axis of x will therefore be cos ^ds, and this being the dif- ferential element of the co-ordinate x, we have dx = cos (pds = cos f' (^) (f^, by (35). Hence x— I cos^f {(j>) dcf) (36), and by symmetry 3/=j'sin^/(^)# (37). Between these equations after integration (^ must be eliminated; the result involving cc, y and two arbitrary constants will be the equation required. It is worth while to notice that the above result may be obtained independently of the consideration of a projection. For since s= Hi + l-^j Y" dx, we have /I ■+(i)"r^-=/<«. ART. 11.] INTRINSIC EQUATION OP A CURVE. 2C.J w!'«"ce \^l + (^J^^'dx=f{4>)d6 (38). But, since -:- = tan cj), the above becomes sec (j)dx =f {(f)) d(j), dx = cos <^/' ((/)) d(^, X = I cos (pf {(j)) d(f>, and In like manner employing for s tlie equivalent formula ^=jsin(^/(<^)#, which agree with the previous expressions. xlnother consequence should also be noted. From ("38) we /■ /-7. . 2i 1 .7 J {-©■jWw ax d'y But T^ = -7- tan"^ [-Y-] = ^T— 7 , whence ax ax \axj f^fyX "^ \d~x) dxJ i + ft ax Therefore i ~ =f' (^). rZo?'^ Now the first member being the expression for the radius of curvature p of the given curve, we have P=/'W (39). Tlius the radius of curvature is determined. 266 INTRINSIC EQUATION OF A CURVE. [CH. XI. 12. Given the ordinary, to deduce tlte intrinsic equation of a curve. The values of s and <^ having been first expressed in terms of the co-ordinates, it only remains to eliminate those co- ordinates between the two equations thus formed and the equation given. Ex. To determine the intrinsic equation of the e;"|ui- angular spiral. The polar equation of the curve being r = (7e"'^, tlie arc s beginning from ^ = 0 is, by ordinary integration, found to be m Again, as the curve cuts all its radii at the same angles the deflection of the arc between two radii vectores is equal to the angle between the radii themselves. Hence the deflection of the arc beginning with ^ = 0 is measured by Q. Therefore 4> = 0, and the intrinsic equation becomes m From this it appears that any intrinsic equation of the form s = a(e"^'^-l) (40) will represent an equiangular spiral. Given the intrinsic equation of a curve, to deduce that of its evolute. Considering the given curve as formed by the unwinding of a string from its evolute, any arc of the former may be said to correspond to that arc of the latter by the unwinding of the string from which it is formed. Thus if s , (/>' represent ele- ments of the evolute corresponding to 5, <^ in the given curve, tlien the origin of s is that point of the evolute whose tangent forms the radius of curvature at the origin of s. This premised, it is evident that we shall have ART. 12.] INTRINSIC EQUATION OF A CURVE. 207 For the extreme differential elements of the arc of the evoliite are respectively perpendicular to the corresponding^ extreme differential elements of an arc of the given curve. Hence the inclination of the former being equal to that of the latter, the value of cf> is the same for both. Secondly, any arc of the evolute is by a known property equal to the difference of the radii of curvature of the ex- tremities of the corresponding arc of the given curve. Hence if p^ represent the radius of curvature at the origin of the given curve, we shall have «' = P-P.=/'( V(39), and, substituting (j)' for cj), «'=/'(f)-/(o). Dropping the accents, we may therefore afHrm that if the intrinsic equation of a curve is s=f{(j)), that of its evolute willbes=/(<^)-/(0). Ex. The intrinsic equation of the logarithmic spiral is s = a (e""'* - 1). Hence that of its evolute is s = mae^'^ — ma which also denotes a logarithmic spiral. Given the intrinsic equation of a curve in the form s=f((f)) wherein /((/)) vanishing with (j> is supposed capable of expan- sion in the form f{^)=A^6 + A^f + A^cl>' + &c (41), required the general intrinsic equation of the involute. As to any curve there belong an infinite number of invo- lutes depending on the different values given to that initial tangent to the curve which forms the initial radius of curva- ture of the involute, we shall represent the arbitrary value of that initial tangent by C. Now if s = F{^) be the intrinsic equation of the involute, we have by the last proposition F'{.t>)-F'{0) =/{). 288 INTRINSIC EQUATION OF A CURVE. [CH. XI. But i^'(O), being the initial radius of curvature of tlie involute, is equal to C. Hence the above equation may be expressed in the form whence F {<}>) = [f{^) d+Cj>+ C\ A o Hence F{ff)) vanishing vv^ith cj), we must have C = 0. Thus the intrinsic equation of the involute, under the condition that its initial radius of curvature is a, will be s=jf{4>)d + a4> (42). If, for distinction's sake, we represent the arc of the invo- lute bj s', the equation may be expressed in the form s'=l{a + s)d(l> (43). It is to be remembered that the lower limit of the integral is 0. The following proposition from the memoir of Hr Whewell referred to, will illustrate the application of the above theo- rems. Let any curve be evolved, and the involute evolved, and the involute of that evolved, beginning each evolution from the commencement of the curve last formed, and witli a " rec- tilineal tail" which is of constant length for all. The curves tend continually to the form of the equiangular spiral. Let s, s', s", &c. be the successive curves, (/> the angle which is the same for all, and let the tails represented in fig. 3, by AA', A' A", A" A'", &G. be each equal to a. ART. 12.] EXERCISES. 269 Then representing the equation of the given curve by s=f[(j))^ we have for the first involute the equation s' = I (a + s) d(j) = «(;£)+ |/(^) cZ^, s" =j{a + s') dcp = acj, + ^,^+ jjf{(p) df, s"' = j{a + s")dcj> = a^ + ^^+j^ + jjjf{f)dc}>\ and in general Now giving to /((/)) the form (41), we have //{<^) #" .n+2 1.2...(7i + l)^1.2...(7i + 2)^ We see then that the first 7i terms of the expression for s'"' in terms of are unaffected by the form of the function /(), while those which remain are affected with coefficients which tend to 0. Thus the limiting form of (44) becomes = a{ef-l) (45). Now this is the equation of an equiangular spiral. EXERCISES. 1. Determine the curve whose subtangent varies as tlie abscissa. 2. Determine the curve whose normal varies as the square of the ordinate. 3. Shew that the curve in which the radius of curvature varies as the cube of the normal is a conic section. 270 EXERCISES. [CH. XI. 4. Find a curve in which the length of the arc is in a constant ratio to the intercept cut off by the tangent from the axis of X, 5. Shew that the above is a particular case of curves of pursuit. 6. Find the orthogonal trajectory of a system of circles touching a given straight line in a given point. 7. Find the orthogonal trajectory of the system of ellipses defined by the equation -2 + y^ = 1, h being the variable parameter. 8. Find the equation referred to polar co-ordinates of the curve in which the radius vector is equal to m times the length of the portion of the tangent intercepted between the point of contact and a straight line drawn from the pole to meet the tangent at a given angle. 9. Eequired the form of a pendent in Gothic architecture supposed to be a solid of revolution, such that the weight to be supported by each horizontal section shall be proportional to the area of that section, 10. Eequired the curve in which 5 = ax^, 11. A curve is defined by this property; viz. that the radius of curvature at any point is a given multiple (n) of the portion of the normal intercepted between the point and the axis of abscissa?; prove that the length of any portion of the curve may be finitely expressed in terms of the ordinates of its extremities. {Cambridge Problems, 1849.) 12. Find a differential equation of the first order of the curve whose radius of curvature is equal to n times the nor- mal, and shew that this is always integrable in finite terms if n be an integer. 13. Shew that if w = 2 the cm've is a cycloid, \i n — 1 a circle, ii. n — — l a catenaiy. 14. The curve whose polar equation is r^ cos mO = oT' rolls on a fixed straight line. Assuming that straight line as the : DIFFEBESTIAL EqUATIONS CH. XI.] EXERCISES. 271 axis of X, shew that the locus of tlie curve described by tlie pole of the rolling curve will have for its equation (Frenet, Recueil d'Exercices sur le Calcul Infinitesimal) Note. To solve problems like the above, we observe that if RTS, Fi^r. 4, represent the given curve rolling on the given line OX, and A PC the curve described by the pole P, then taking OX for the axis of x, and putting OM=x, MP — II, the straight line PT joining that pole with the point of contact will be a radius vector of the given curve, but a normal of the described curve. Hence ^-^^hm\ <«>• Again, PM is the pei-pendicular let fall from the pole upon the tangent of the given curve, but the ordinate y of the required cm-ve. Hence =y (^). By means of (a), (5), and the equation of the given curve, eliminating r and 6, we obtain the differential equation of the curve sought. 15. In the particular case oim = \ the rolling curve will be a parabola, the pole its focus, and the described curve a catenary. 16. If m = 2, the rolling curve is an equilateral hyperbola, the pole its centre, and the described curve an elastica. ( 272 ) [CH. xir. CHAPTER XII. ORDINAEY DIFFERENTIAL EQUATIONS WITH MORE THAN TWO VARIABLES. 1. The class of equations wliicli we shall first consider in tliis Chapter, is represented by the typical form, Pdx+ Qdy + Rdz = 0 (1), P, Q and R being functions of the variables x, y, z) and it is usually termed a total differential equation of the first order with three variables. Possibly the first observation suggested by the examination of this form will be, that it does not answer to the definition of a differential equation, as the expression of a relation in- volving differential coefiicients. Chap. I. And certainly it does not exhibit their notation. If, however, we attempt to attach a meaning to the general form (1), we shall perceive that the idea of a limit is involved essentially. And if we study its origin, we shall see that this idea may be expressed, here as elsewhere, in the language of differential coefficients. For (1) is not understood as implying simply that the expression, FAx+QAy + BAz (2), approaches to the value 0 when the increments Ax, Ay, Az approach that value, true though it be that the vanishing of the increments causes that expression to vanish with them. But what (1) is always understood to express is, that in tlie approach to the limiting state, (2) tends to vanish in conse- quence of the ratios which the increments Ax, Ay, Az tend to assume; it is, that if we represent (2) in any of the equivalent forms FAx + QAy + BAz . PAx + QAy +RAz ^ „ Ax ^^' Ay' ^^'^^- the limit of the ratio expressed by the first factor of each is 0. And the problem of the integration of (1), is that of the discovery AIIT. 1.] EQUATIONS WITH MORE THAN TWO VARIABLES. 273 of the possible relation or relations among the primitive vari- ables which will secure this result, supposing Aa^, A^/, A.-j to be so restricted as to preserve such relations unviolated. Now whether the primitive variables are connected by one equation or by two simultaneous equations (we cannot\sup- pose them connected by three equations without making them cease to be variable), the relation (1) is fully expressible in the language of differential coefficients. If there exist one primi- tive rehation which, as we shall hereafter see, can only happen under particular circumstances, then , dz ^ dz ^ dz — -r- dx + -J- dy, ax dif "' while (1) is presentable in the form dz = — ^ dx — -P7 dy. Hence, since dx and dy are independent, we have dx~ R' dy~ R (^'' a system which in the supposed case is equivalent to (1). On the other hand if, as Avill usually happen, two simultaneous equations connect the primitive variables, e.g, ^{^,y^ ^)=0, ir(^x,7/, z)=^0 (4),- then since we have d(h dd> -, d6 ^ -j^dx+ ---di/ + -fdz = 0, dx dij ^ dz ' V^ dx -^^-i^ dy + ^J dz = 0, dx d'j ^ dz the elimination of dx^ dy, dz between these and the original equation gives /d(^ dyjr _ rZ (z) dz fjL {Pdx + Qdy) + ]-^ ^[ dz = 0. ART. 4.] AVITII MOllE THAN TWO VARIABLES. 277 Now hy the given equation, PJx+ Qchj = — Piclz. SiiLsti- tuting, and rejecting the common factor dz^ we have ^ dV dcf>(z) ^ whence *#=?-"= ('■). the second member of which must, on tlie hypothesis that a single primitive exists, be reducible to a function of z by means of (10). The solution of the equation thus reduced will determine <^ (2;), the value of which substituted in (10) will give the complete primitive. Although we are fully entitled to affirm that the equation determining {z) must, whenever a single primitive exists, be reducible to a form not involving x and ?/; it may be pro- per to verify this conclusion a posteriori. Let us then inquire under what condition the function — fjuR^ can be freed from both x and y by means of the equation V= {z). Evidently this can onlv be the case when dV . ' . — /jlE and V are so related that, considered with respect to X and y alone, the one is a function of the other. Thus we have by the equation of condition (Prop. 1. Chap, ir.) ^ A i^- e) -^ — (—- r] = o dx dy \dz ) dy dx \dz ^ J ' or dV drV_ _ dV drV_ fdR d_V_dB d_V\ dx dzdy dj dzdx \dx dy dy dx) ^jt(i^'k_'lI'!t) = o (i^\ \dy dx dx dy I ' ' JNow smce -i— = uR -^— = aO, we liave dx ^ dy ^ 278 ORDINARY DIFFERENTIAL EQUATIONS [CH. XII dV dW dV d'V T.d , ^^ r>d , ry. ^YS.-^'S) (^•^)' Thus also (dRdV dRdV\ ^f^dR ^dR\ ,,,, ^Kdxdy-^-dx)=^[^^-^-dy) (^^)- Lastly, nf£-fS-^('3|-^|) (-)• But since yu. is the integrating factor of Pdx -f Qdy we have by Chap, iv., Q±:_pdf^^ fdP__d_Q\ dx dy \dy dxj ' which reduces (15) to the form j^fdVd^_dVd.^ ,dPdQ-^ \dy dx dx dy) \dy dxJ Substituting these values in (12) and rejecting the common factor /A^, there results dz dz dx dy ay dx or ^(dQ dR\ ^(dR dP\ r>(dP dQ\ ^ ,^. and this is identical with the equation of condition (8). The conclusion is therefore established. It follows also that it is not necessary in any proposed case to apply directly the above equation of condition. It is im- 2}UcitJy involved in the very process of solution. 5. The results of the above investigation are contained in the following Rule. ART. 5.] WITH MORE THAN TWO VARIABLES. 279 Rule. Integrate the j^roposed equation on the hypothesii that one of the variables is constant and its differential therefore equal to 0, adding an arbitrary function of that variable in tlie l^lace of an arbitrary constant. Then differentiating icith respect to all the variables, determine the arbitrary function by the condition that the result of such differentiation shall be equi- valent to the equation given. The equation expressing such con- dition will, if a single jpriniitive exist, be reducible by prevdo^is results to a form in ivhich no other variable than the ona in- volved in the arbitrary function will remain. Ex. 1 . Given {y + o)'^ dx 4- zdy — {y + a) dz = 0. Here P = {y -{. aY, Q = z, R — — y — a, values which identi- cally satisfy the condition (8). The equation therefore admits of a single complete primitive. Regarding z as constant we have first to integrate the equa- tion (j + a)' dx + zdy = 0. Dividing by (?/ + «)^, we have 7 zdii dx+ . ■' ^0, (y + ^y the solution of which is X = 6 iz), y + a ^ ^ ^' (^ [z) being an arbitrary function of z introduced in the place of an arbitrary constant. Now, differentiating with respect to all the variables, we [y + a) ^ \y^a dz ] or {y + af dx -Vzdy - L + a + (^ + rt)' -^y [ dz = 0, which agrees with the equation given, if we have or -^V- = 0. dz 2S0 OEDINAllY DIFFERENTIAL EQUATIONS [CH. XII. Here then 6 (.:•) = c and the complete primitive is X =c (a). If we commence by regarding y as constant we obtain by a first integration z-[y-^a)x=^[ij), whence, differentiating and comparing with the giA^en equa- tion, chj y-\-a' This eo[uation involves both x and y, but it is reducible by the previous one to the form dy y + a ' cl(j) (y) dy or — ^^^= — '' — ^ (7/) 3/ + a ' of which the integral may be expressed in the form h being an arbitrary constant. Hence, finally 2; = (?/ + a) X + 6 (?/ + a) = [y + a)[x + h), and this is equivalent to the former result (a), Ex. 2. Given zdz -\-{x-a) dx = {A' -z''-{x- af]- dy. Integrating as if?/ were constant we have £'+[x-aY=(l>{y) (a). Differentiating and comparing with the given equation 1 d(^ (V) f,o .> , NOli = [V-{y)]i '^y- Anr. C] AviTH Mor.E than two a'ariables. 281 Therefore integrating h being an arbitrary constant. Hence determining 0 (?/), and substituting in (a), we have finally wliere h is arbitrary. Homogeneous Equations. G. "When the equation Fdx + Qdy + Bdz = 0 is liomo.^-e- neous with respect to x, y, z, its solution will be facilitated by a transformation similar to that employed for homogeneous equations Avith two variables. Assuming x ■= uz, y = vz, we obtain by substitution a result of the form d^ L ^^Mdu + Ndv (18). If L be equal to 0 this simply gives Mdu + Xd\^ = 0, which can always be made integrable by a fixctor. If L be not equal to 0 we have dz M -, N ^ — = -^ da -\- -Y dv ] Z h Li and here the first member being an exact differential the second will be such also if a complete primitive exist. After integration, u and v must be replaced by their values -, '- . Ex. 3. Given {ay — hz) dx + [cz — ax) dy+{bx — cy) dz = 0. This equation satisfies the equation of condition (8). Assuming x = uz, y = vz it becomes simply {av—h) die — {ait — c) do = 0, du dv or = J , mc — c av — 0 282 INTEGRATING FACTOES. [CH. XII. the solution of whicli is au— c ^ 7 = ^> av — o whence the complete primitive sought will be ax — cz ^ ay — bz Ex. 4. Given [if + y^ + ^') dx + (^' + xz-\- z^)dy + (x- + xj/ + y-) dz = 0. Assuming x = uz, y = vz, we have on reduction dz _ (i;' + V+1) dit + (?r + u + l)dv z ~ ((a-^v+1) {uv + u + v) ' dz du + dv (v 4- 1) du + [u + 1) c?y or Z U + V +1 uv -\-ic + v v\'hence integrating lo2;^ = lo2^ + (J. = C', Finally we have xy -\- XZ+ yz x + y + ^ for the complete primitive. The last two equations might have been integrated without preliminary transformation. (Lacroix, Tom. Ii. pp. 507 — 510). Integrating factoids. 7. The equation Pdx + Qdy + Edz = 0 can also, when there exists a single complete primitive, be integrated by means of a factor. If [jb be that factor, then, since tlie expression IJiPdx + [Ji Qdy + fiBdz ART. 8.] INTEGRATING FACTORS. 283 must be an exact clifFerential, we must have d{fiQ) ^d(jMR) clifjiR) _d{fj,P) dz dy ^ dx dz ^ d{ixP) _ difiQ) dy dx ' equations to wliicli we may give the forms 4:-«l-(f-S)=»- Multiplying these equations by P, Q, and R, respectively, adding, and dividing by /x, we have '(f-f)*«(S-S)+Mf-S)="-""' the same equation of condition which was before obtained. When this equation is satisfied a particular form of the factor /x will frequently suggest itself. In Ex. 3 the functions (ay - bzf ' (cz - axf ' [hx - cyy are integrating factors. In Ex. 4 the functions and 7 — are inteGrratino; factors. {xy + xz + yzY ° ° {x-\-ij + z)- Equations not derivable from a single j^rinutive. 8. To solve the equation Pdx + Qdy + Rdz = 0, when the equation of condition (8) is not satisfied. 284 EQUATIONS NOT DERIVABLE [CH. XII. Ill this case the solution consists of two simultaneous equa- tions between a?, y, z, one of which is perfectly arbitrary in form. For representing an assumed arbitrary equation in the form f[x,y,z) = 0 (20), and diiferentiatin2\ we have ^/'fe Ih ^) 7 , df{x,y, z) dfix, y. Xow these two equations enabling us, wlien the form of / («^j y^ ^) is specified, to eliminate one of tlie variables and its differential, e.g. z and dz^ from the equation given, permit us to reduce it to the form Mdx 4- Ndy = 0, J/ and N being functions of x and y. • Solving this, we obtain an equation involving an arbitrary constant, and this equation together with (20) will constitute a solution. By giving dif- ferent forms to f{x, y, z) every possible solution may be ob- tained. AVhat a solution thus found represents in geometrical construction is the drawing, on a particular surface, of a family of lines, each of which satisfies at every point the con- dition Pdx+ Qdy + Bdz = 0. Now dx, dy, dz are propor- tional to the directing cosines of the tangent line. Hence the geometrical problem may be represented as that of drawing on a given surface a family of lines, in each of which the directing cosines cos , cos yjr, cos % at any point shall satisfy the con- dition Pcos(/)+ () cos i/r + i? cos % = 0 (21). Ex. Required the most general solution of the equation xdx + ydy + c(^l-^,-^Jdz = 0 (a), which is consistent with the assumption that it shall represent a series of lines traced upon the ellipsoid whose equation is x^ f z'' ART. 8.] FROM A SINGLE PRIMITIVE. 285 It will be found tliat (a) docs not satisfy the equation of condition (8). Differentiating (5), we have xdx ydy zch a^ 6" c^ , , c^ fxdx ydy whence dz = — „ + ^ '^ z a^ ' h fxdx ydy X- 7/-^^ 1- a ■2 J2 and this reduces (a) to ../. + y.?,-c=(^+^)=0 (c), the integral of which is indicating that the projections of the proposed family of lines will be a certain series of central conic sections. l^a = h = c = l the proposed equation admits of a single primitive, viz. x^ -\- y'^ + z^=\. And any line traced on the surface of which this is the equation will satisfy the differen- tial equation ; for the equation (c) by which the lines are ordinarily determined is now reduced to an identity. The above method of solution is due to Xewton. Monge has however remarked that the general solution may be ex- pressed by the equations (10) and (U) of xVrt. 4, viz. by the simultaneous system V=(z) (22), ^-^^R = i>■{z) (23), where /jl is the integrating factor, and V the corresponding integral of the expression Fdx + Qdy. It is indeed shewn iu 28G DIFFERENTIAL EQUATIONS CONTAININa [CH. XII. that Article that (22) does satisfy the differential equation provided that the condition (23) is satisfied. But there is no 2)ractical advantage in the employment of Monge's form. Applied to the problem of drawing on a given surface lines satisfying the condition expressed by the differential equation, it makes the determination of the arbitrary function (^ [z) itself dependent on the solution of a differential equation. Thus in the example last considered we have, on giving to fjL the value 2, so that the general solution assumes the form To a])ply this to the problem of drawing lines satisfying the conditions of the problem on the ellipsoid X' 7f Z' ^ ,' ^^¥-^? = ' (^)' it is necessary from the above three equations to eliminate x and y. From the second and third which here suffice, we have whence (\> {z) = — z^ + C. Therefore x" ^ y^^ j^ z"" =^ C (/). The particular solution sought is therefore expressed by the equations (e) and (/), which are together equivalent to the previous solution expressed by {h) and {d). Total differential equations containing more than three va- riables. 9. It will suffice to make a few observations on the equa- tion with four variables Pdx-\- Qdy + Rdz+ Tdt = 0 (24), and to direct attention to the general analogy. ART. 9.] MORE THAN THREE VARIABLES. 287 Writing the above equation in the form P , Q , R , dt = - -fj,dx --jj,cbj - -j,dz ( 2 0 ) , it is evident that, in order that it should be derivable from a single primitive, we must have \dx) T~[dii)T' [dii)T~[dz)T' \dzjT~\dx T' where f -j- J refers to x not only as appearing independently, but also as implicitly involved in f ; and so on for the rest. Effecting the differentiations, and substituting for -f •> ~r ^ y their values implied in (25), we have U.C dy)^ \dy dlj^^[dt dx) I \d^i/ dz J ^ \dz dt) \dt dij J ^ ^ ' ^(dP dR\ ^[dT dP\ ^(dR dT\ ^ which arc the equations of condition of existence of a single complete primitive. It is evident from the symmetry of the problem that the equation ^(f-f)-«(S-'S-''(f-S)=«-<"' must also hold here. But this is not a new condition. It may be deduced from (2G), by multiplying the respective equations of that system by R, P, and Q, and adding the results. 2S8 EQUATIONS WITH MORE THAN THREE VARIABLES. [CH. XII. It is obvious that when there exist n variables, the number of independent equations of condition is -~ — , being the number of ways of equating two partial differential coefficients in a system in which n— 1 are contained. The solution of any such equation may be effected by an extension of the method adopted for equations with three variables. We must integrate as if all but two of the varia- bles were constant, adding, in the place of an arbitrary con- stant, an arbitrary function of the variables which remain. This function we must determine by differentiating with re- spect to all the variables, and comparing with the equation given. If a single primitive exist, such determination will be possible. If a single primitive do not exist, we must, follow- ing the analogy of the corresponding case for three variables, endeavour to express the solution by a system of simultaneous equations. And such is indeed its general form. Pfaff, in a memoir published by the Berlin Academy 1814 — 15, has shewn that, according as the number of variables is 2n or 2r + 1, the number of integral equations is n or n-\-l at most. His method, which is remarkable, consists of alternate inte- grations and transformations. For important commentaries and additions see Jacobi {WerJce, Tom. I. p. 140), and Eaabe {Crelle, Tom. xiv. p. 123). Ex. Given {2x + 3/^+ ^xt/^—t/^) dx + Ixy dij—xdy^ + x'dy,^ = 0. If we suppose the variables t/^, y^? constant, we have to in- {^x^y- -1- 2a'?/2 — y^ dx+ Ixydy = 0, which, on substituting an arbitrary function of ?/j ,?/„ repre- sented by (f), for an arbitrary constant, gives X" + xy^ + x^y., — xy^^ = cf). Differentiating with respect to all the variables, we have {2x + ^" + 2xy.2 — yj dx + 2xydy — xdy^ + x^dy.^ dy^ -^^ dy, ^^ tegrate ART. 10.] EQUATIONS WITH MORE THAN THREE VARIABLES. 289 Comparing this with the given equation, we have whence cj) = c and the solution is x^ + X7/^ + x^i/2 — xy^ =c (a). Had we begun bj making x and y constant, we should have had as the result of the first integration, ^y-2-^y, = W- (^ denoting a function of x and y. Differentiating with respect to all the variables and comparing with the given equation, we should find d(f> = — {2x + y^) dx — 2xy dy, whence, (f) = — x^— xy^ + c, the substitution of which in {b) reproduces the former solu- tion (a). Equations of an order higher than the first. 10. When an equation of the form Adx'-\-Bdy'+ Cdz'-[-2Ddydz + 2Edxdz-{-2Fdxdy=:0...{2S), is resolvable into two equations each of the form Pdx + Qdy + Fidz = 0, the solution of either of these obtained by previous methods, will be a particular solution of (28), and the two solutions taken disjunctively will constitute the complete solution, which is therefore expressed by the i^roduct of the equations of these solutions, each reduced to the form F= 0. The condition under which (28) is resolvable as above, is expj'essed by the equation, ABC^2DEF-AD'-BE''- CF'=0 (20). C.D.E. Id 290 EQUATIONS OF A HIGHER OEDER. [CH. XII. This is sliewn hy solving (28) witli respect to dx, and assuming the quantity under the radical to be a complete square. Thus, the equation x^dx"^ + y^d]f- — z^dz^ + 2xydxdy = 0, which will be found to satisfy the above condition, is resolv- able into the two equations, xdx + ydy + zdz = 0, xdx + ydy — zdz — 0, whence, ic^ + 3/^ + 2;^= c ... (a), cc^4-y-s^ = c' (J). Geometrically the solution is expressed by lines drawn in any manner on the surface, either of the sphere (a), or of the hyperboloid (Jj), When the condition (29) is not satisfied, the proposed equation does not admit of a single primitive, or of any dis- junctive system of primitives. But it does in general admit of a solution expressed by a system of simultaneous equations. Thus, if we integrate the equation dz^ = nt^ {dx^ -^ dy^) , sup- posing x constant, we find z = my + (7, or, replacing (7 by a function of a?, z = my + <^(^) y{c). On substitution and integration, we find that this will satisfy the proposed equation if we have 2y = ^>? TTTT (f>{x)-\-c {d), "^ J

{x, y, z) =c '? 11. Shew that any system of lines described on the surface of the sphere x^ + y^ + z^ = r'^, and satisfying the above equa- tion, would be projected on the plane xy in parabolas. 12. Shew that Monge's method would, if we integrate first with respect to x and z, present the solution of the equa- tion of Ex. 10, in the form \l + 2m)x' + z' = cl,(y), 2y {1 - x) = - cj,' (y) . 13. Applying this form to the problem of Ex. 11, form and solve the differential equation for the determination of (f) {y), and shew that it leads to the result stated in that Ex- ample. 14. Find the equation of the projections of the same system of curves on the plane yz, 19-2 ( 292 ) [CH. xiii. CHAPTER XIII. SIMULTANEOUS DIFFERENTIAL EQUATIONS. 1. We have hitherto considered only single diiferential enuations. We have now to treat of systems of differential equations. Of such by far the most important class is that in which one of the variables is independent and the others are depend- ent upon it, the number of equations in the system being equal to the number of dependent variables. Thus in the chief problem of physical astronomy — the problem of the motion of a system of material bodies abandoned to their mutual attractions — there is but one independent variable, the time ; the dependent variables are the co-ordinates, which, varying with the time, determine the varying positions of the several members of the material system ; while, lastly, the number of equations being equal to the number of co-ordinates involved, the dependence of the latter upon the time is made determinate. Such a system of equations may properly be called a deter- minate system. We propose in this Chapter to treat only of systems of equations of the above class. And in the first instance we shall speak of simultaneous differential equations of the first order and degree, beginning with particular examples, and proceeding to the consideration of their general theory. Particular Illustrations. 2. The simplest class of examples is that in which the equations of the given system are separately integrable. Ex. 1. Given Idx + mdy -\- ndz = 0, xdx + ydy + zdz = 0. Integrating separately, we have Ix + my + nz = c, x^ + y^ + z'^ = c' ; and these equations expressing the complete solution of the given system may be said to constitute the primitive system. ART. 3.] PARTICULAR ILLUSTRATIONS. 293 Another class of examples is that in which, wliile the equa- tions of tlie given system are not all separately inte,a:rable, they admit of being so combined as to produce an erjuivalent system of equations which are separately integrable. Ex. 2. Given -^ + _ = 1, -^/ = ^ +^ + -_ _ 1. Here the first equation alone is separately integrable, and gives t c , . ^ = 3+^^ (<')• Also by addition of the given equations, we have dx + dy dt = x + y therefore = dt, X+7J log{x + i/) =t-\-c' [h). The primitive system is therefore expressed by (a) and {h). In both the above examples we see that the number of equations of the solution is equal to that of the equations of the system given, and that each equation of the solution in- volves a distinct arbitrary constant. And it is evident that this must be the case whenever we can combine the given equations into an equivalent system of integrable equations of the first order. But as we have not proved that such combi- nation is possible, the following question becomes important. viz. what is the nature of the solution of a system of simulta- neous equations of the first order and degree ? This question will be considered in the next section. General theory of simultaneous equations of the first order and degree, 3. We shall seek first to establish the general theory of a system composed of two equations between three variables, and therefore of the form Pdx-^ Qdy + Bdz^O, } . P'dx+Q'dy-{-Il'dz = 0,] ^ ^' 294 SIMULTANEOUS DIFFEKENTIAL EQUATIONS. [CH, XIIT. the coefficients P, P', &c. being functions of the variables, or constants. We design to consider the above system first, and with the greater care, because there is scarcely any part of the general theory which it does not serve to exemplify. Peop. The solution of the system (1) can always he made to depend upon that of an ordinary differential equation of the second order hetween two of the primitive variables, and it always consists of two equations involving tivo arbitrary constants. By algebraic solution of the system (1) we have , RF-PR' -. PQ'-QP' ,^. ^y^-QE^RQ^''^ ^''=QP7^RQ'^'" ^'^* As the coefficients of dx in the second members of these equations are functions of x, y, z we may express the reduced system in the form dy = (f) {x, y, z) dx^ dz = '^ {x, y, z) dx, whence, regarding x as independent variable, % = 4>{^,y,z) (3), dz Thus the given system enables us to express -^ and -^ by known functions of x, y, z. Now differentiating (3), still on the assumption that x is the independent variable and representing for brevity (f> {x, y, z) by <^, '^ (a?, y, z) by -v^, we have d'^y _ d(f> d(f> dy d(j) dz dx^ dx dy dx dz dx ' dz or substituting for -,- its value given by (4), ^^#_^i^^_^ . # .^. dx^ dx dy dx ^ dz ART. 4.] PARTICULAR ILLUSTRATIONS. 295 7 ^2 This equation involves -j- and -^^ together with the quan- tities -y- , -y-, -j^ and yjr, which arc known functions of x, y, and z. Hence eliminating z by means of (3) we have a linal equation involving -j^- , ^ , x, and y. The complete primi- tive of this differential equation of the second order will enable us to express 3/ as a function of x and two arbitrary constants. Suppose the value thus obtained for y to be ?/ = X(a^, c,,cj (6). Then we have by virtue of (3) (i>{x,y,z)= ^' ^J '^ (7). These two equations involving two arbitrary constants con- tain the complete solution of the system given. 4. It is important to observe that the system (2) may be expressed in the symmetrical form dx _ dy dz QR -R(y~ RF^ PE " PQ - QF ' If we represent the denominators of the above reduced system by X, Y, Z, it becomes dx _dy _ dz x-y-z ^^"• This, then, may be regarded as the symmetrical form of a system composed of two differential equations of the first order. Again, the complete solution of such a system, as is expressed by (6) and (7), consists of two equations connecting the varia- bles x, y, z with two arbitrary constants. If we solve these equations with respect to the constants, the solution assumes the form i(^>2/. ^)=c„ -J = (a + 77ia ) x+ io + mo ) 7/ + c + 7nc ctt , ,. ( h + mh' c ■\-mc = (a -1- ma )\x-\ — -, y + a -{-ma ^ a + ma = (a + ma ) x + my -\ ~, [a), ^ ^ ^ ^ a-\- ma / ART. 7.] WITH CONSTANT COEFFICIENTS. 301 provided that we determine m so as to satisfy the condition [b). h + mb' m = — ; -, a + ma > or a'm^ + (a — b') m — b-- = 0 . Now («) gives dx + ondy ^ 5 + ma] )dt, c + 7nc ^ X + w?/ + , a + ma whence on integration , / c-\-mc\ loo- X 4- mij A : (a^ mo.') t 4- n .... In this equation it only remains to substitute in succession the two values of m furnished by (b). The two resulting equations, in which the arbitrary constants must of course be supposed different, will express the complete solution of the problem. When the values of m are equal, the form {c) furnishes directly only a single equation of the complete solution. We may deduce the other equation, either by the method of limits (assuming the law of continuity), or by eliminating x from the given system by means of (c), and then forming a new differential equation between y and t. It seems preferable however to employ the general method of Art. 5, by which all difficulties connected with the presence of equal or imagi- nary roots are referred to the corresponding cases of ordinary differential equations. 7. Simultaneous equations are so often presented under the symmetrical form (11) that the appropriate mode of treatment deserves to be carefully studied, especially as it possesses the superiority, always in point of elegance, and frequently in pohit of convenience, over other processes. It is known that eacli member of a system of equal frac- tions is equal to the fraction wdiich would be formed by divid- 302 LINEAR EQUATIONS OF FIRST ORDER [CH. XIII. ing any linear homogeneous function of their numerators by the same function of their denominators. Hence if we have a system of equations of the form X, ~ X, ~ X,~ T ^ ^^' in which we suppose t the independent variable, and T a function of t only, then we shall have dt _ dx^ + mdx^ . . . + rdx^ /. -x T~ X, + mX,...+rX, ^ ^' Hence, should the first member be an exact differential, the inquiry is suggested whether the multipliers m,...r cannot be so determined, whether as functions of the variables or as constants, as to render the second member such also. Now wdien the system of equations is linear and with constant co- efficients this can always be effected. It may be observed that the cliaracter of the system is as manifest from inspection of the symmetrical form (16) as of the ordinary form. If the system be linear and with constant coefficients the denomina- tors X^, X^^.,.Xn will, when considered with respect to the dependent variables x^, x^^...x^^ be linear and with constant coefficients. In the employment of this method it is often of great ad- vantage to introduce a new independent variable, and to con- sider all the variables of the given system as dependent upon it. We are thus enabled to secure the condition above adverted to, of having one member of the symmetrical system an exact differential. 17 ^. dx dy JliX. (jriven ax-\-hy-\-G dx + h'y + c' ' Let us introduce a new variable t so as to give to the system the form dx dy dt , s —- f, - = — (a). ax-{- by + c ax + oy -{- c t ART. 7.] WITH CONSTANT COEFFICIENTS. 303 Here the tlilrd member being an exact differential, we shall write dt _ dx + mdy t ax + b7/ + c + m {ax + J'y + c) _ dx-\- rtidy [a + ma) re + (^ + mh') y + c + mc _ 1 {a + ma) dx + (a + ma) mdy a + ma {a + md) x-\- {b + mb') y + c + mc ' The second member of this equation will be an exact differen- tial if we have {a + md) m = b-\- mb' (Jj)^ the integral corresponding to each value of m thus determined being of the form log ^ + (7 = — ■ -, log {(a + md) x-\-{b + mb') ?/ + c + mc'], ^ a + ma ^ ^ ' '^ ^ ' or Ct = {aa; + 5^ + c + m {ax + Vy + c')}"^^'. If the roots of the quadratic i^) are m^ and m^, we thus find C^t =[ax + by-\-c-\- m^ {dx + b'y + c'^1"^'"''' C.J. = {ax + by-\-c + m^ {dx + h'y + c')] ' ' (o), 1 I a+m.,n for the primitive equations of the system (a). Tliose of tlie given system will be obtained by eliminating t. The result assumes the remarkable form {ax + by + c-i- m^ {dx + b'y + e) l"^'"'"' ^ , ,^ {ax + by + c + m^ {dx + b'y + c)}"^'''' Ex. 2. Given -^ = ^ = -^, where X= ax + by 4- cz + d \ Y=dx-\-b'y-i-cz + d' > (a). Z = a"x+b"y+c"z-hd" J y w- 304 LINEAR EQUATIONS OF FIRST ORDER [CH. XIII. Introducing a new variable t, so as to give to the system the more complete form dt dx dy dz .,, 7=x=r=^ (*)= , dt Idx + tndy + ndz we have -- = —tt- r^r — ^ _ Idx + mdy + wd'^ . . X {Ix + my + 7iz + r) Provided that we assume al + am + a"n =\l ^ hi + &'?;i + h"n = \m cl + cm + c"n = \n dl + c^'m -\-d"n = X?' i The first three of these may be written in the form (a — X) 1 + am + a"n = 0\ hl + {h'-\)m +h"n = o[ (e), cl + c'7n + [c" — \)n = 0} Avhence eliminating I, m, n we have the well-known cubic {a - X) {h' - X) (c" - X) - h"c {a - X) — ca" ih' — X) — ha (c" — X) + dh"c + ci'hc =0 ... (/). Now let the values of X hence found be X^, \^ \^ and the corresponding values of I, m, n, r, be l^, m^, n^^ 7\, l^, tn^, &c. then integrating (c) we shall have the system i_ c,t = [l^x + m^ij + n^z + rj^i, c^t = {l^x + m^ + n^z + r J^-', Hence eliminating t by equating its values, we find as the general solution of the original system of equations ART. 8.] WITH CONSTANT COEFFICIENTS. 305 [\x + m^rj + n^z + rj^^ = C {I^x + m^y + n,^z + rj^-^ i_ = C {I^x + m^y + n^z + 7-3)^3 . . . ^rj). In the same way we may integrate the general system dx^ _ dx^ _ dx^ X^ X^ X„ ' where Xj, X2,...A\, are any linear functions of the variaLles. 8. From the ahove results tlie solutions of various sym- metrical systems in which the denominators are not linear may be deduced. The most remarkable of such deductions is the following. Suppose that in the system dx _ dy _ dz ax + hy + cz ax + h'y + cz ax + h'y + c' z' " * ^ the solution of which is known from what precedes, we sub- stitute X = xz, y = yz\ X and y being new variables introduced in the place of x and y\ The result is zdx + xdz _ zdy + ydz _ dz ax + hy + c a'x + h'y + c a"x + h"y + c" ' to which we may obviously give the form zdx zdy ax-\-by-\-c—x (d'x + b"y + c") dx-\-h'y ■\- c —y [a'x ■\-b''y-\- c") dz d'x + h"y + c" * Dividing the first equation of this system by s', we have ^ ^ (^Jl ^ /n^ ax+by-\-c—x {a" x+b"y-\-c") dx-\-b'y-\-c—y {a"x+b"y+c") Now this on clearing effractions will be found to be of the same form as Jacobi's equation {Crelle, Tom. xxiv. p. 1), whose solution on other grounds has been explained, Chap. v. Art. 8. B. D. E. 20 306 LINEAK EQUATIONS OF FIRST ORDER [CH. XIII. We see that the solution of (h) is deducible from that of the system («) by changing x into xz', y into yz ^ and elimi- nating z . And just in this way the solution of any symmetrical non-linear system of the form ^^\ ^*^2 dx^ , . X^ — x^X X^ — x^X " X, — x^X in which X, X^, X^,...X„ are linear functions of the variables x^, x,^,...Xn may be made to flow from that of a symmetrical system of the form in which Xj, X^,...Xn_^_^ are linear homogeneous functions of the variables ic^, x,^,...Xn+^. The general solution of the sys- tem (18) seems to have been first obtained by Hesse {Crelle, Tom. XXV. p. 171). 9. Lastly, certain systems of linear equations which have not constant coefficients may be solved by the above method. Thus the solution of the equations («), ^+T(ax + b^)=T, where T, T^, T^ are functions of the independent variable, may be reduced to that of an ordinary linear differential equa- tion of the first order. For proceeding as before, we find ^+J^) +y,T[x + my) = r, + mi; [b), provided that \ and m be determined by the conditions X = a + ??ia', \m = h + mh' (c). ART. 10.] WITH CONSTANT COEFFICIENTS. 307 Hence eliminating \, we have m {a-^ma) =h-\-mV [d), which gives two values for m. Integrating [h) regarded as a linear equation of the first order between x + my and t, and substituting for \ its value in terms of m given by the first equation of the system (c), we have x+rmj = e-^^^'^y^' { G + /e^^-^^-V ^^ {T^ + m T^ dt] (e) , in which it remains to substitute for m its values given by {d). ^ n. dx 2 , . ^ dy 1 , ^ . Ex. Given ^ + -{x-7j) = l, ^^+-{x + oy) =L The solution is If in the system (a) we make T=l, it becomes a system of equations with constant coefficients but possessed of second members. The general system analogous to (a) when the number of variables is increased, may be solved by the same method. It may be well to notice that the equivalent symmetrical form is (20), where X^, X^,...Xn are linear homogeneous functions of the dependent variables, and T, T^^...T^ are functions of t. Treated under this form, it is obvious that its solution will be made to depend upon that of a linear differential equation of the first order, and an auxiliary algebraic equation of the n^^ degree. Equations of an order higher than the first, 10. Any system of simultaneous equations of an order higher than the first is reducible to a system of the first order. 20—2 dx^ dx^ dx^ dt X^+T^ X^+T^" ' X^+T^ T 30S EQUATIONS OF AN ORDER [ciI. XIII. And this reduction though not always necessary for the pur- pose of solution is theoretically important, because it enables us to predicate what hind of solution is possible. To eiFect this reduction it is only necessary to regard as a new variable and to express as such by a new symbol, each differential coefficient, except the highest, of each dependent variable in the given equations. The transformed equations will thus be of the first order, and the connecting relations of the first order also; and the two together will constitute a system of simultaneous equations of the first order. Ex. Given the dynamical system d^'x de ~ ■x, df ~ ' d'z df^^' where X, Y,ZaYQ functions of the variables. Here if we assume dx dt'~ = x, dt ^' dz , dt='' the given system assumes the form dx dt = x, dl_ dt -*' dt Thus we have in the whole six equations of the first order between the six dependent variables x, y» z, x\ y\ z\ and the independent variable t. The complete solution of the latter system will therefore consist of six equations connecting the above system of varia- bles with six arbitrary constants. If from these six equations we eliminate the three new variables ic', ?/', z\ we obtain three equations connecting the original variables cc, ?/, 0, t with the above-mentioned six arbitrary constants. And thus it might be shewn that the complete solution of any system of three differential equations of the second order between four variables will be expressed by three primitive equations connecting these variables with six arbitrary con- stants. ART. 10.] HIGHER THAN THE FIRST. 309 And still more generalhj, the comi^lete solution of a system of n differential equations containing n+\ variables of ic hick one is independent will consist of n equations connecting those rariahles with a number of constants equal to the sum of the indices of order of the several highest differential coefficients. For let t be the independent and x one of the dependent variables, and let the highest differential coefficient of x d''x which presents itself be -v^ . Tlien in the reduction of the system of given equations to a system of equations of the first ( srder it is necessary to introduce n — 1 new variables con- nected with X by the relations dx dx^ dx,,. dt'"^'' dt ~'^''"' dt ~''"-^- Thus the number of variables in the transformed system cor- responding to X and its differential coefficients will be n, and as a similar remark applies to all the other variables, it ap- pears that the total number of variables of the transformed system will be equal to the sum of the indices of the orders of the highest differential coefficients of the several dependent variables in the system given. Such then will be the number of equations of the transformed system, and such the number of constants introduced by their complete integration. Art. 5. It is also evident that if from the equations by which the complete solution is expressed we eliminate all the new variables there will remain a number of equations equal in number to the original equations, and connecting the primi- tive variables with the constants above mentioned. Thus the proposition is established. The transformation above employed is further important, because in the highest class of researches on theoretical dy- namics it is always supposed that the differential equations of motion are reduced to a system of simultaneous equations of the first order. At the same time it is not necessary for ordinary purposes to effect this reduction. Difterentiation and elimination al- ways enable us to arrive at a differential equation, higher in order, between two of the variables. The method of indeter- ;>10 EQUATIONS OF AN ORDER [CH. XIII. minate multipliers maj also be sometimes used with advan- tage. No general rule can however be given. [The statement respecting the number of arbitrary constants is not universally true. Suppose, for example, that there are two simultaneous differential equations which connect x and y with the independent variable t. Let one equation contain differential coefficients up to ^ - and -^ inclusive ; and let d^'x the other equation contain differential coefficients up to -j-^ d^ii and -^ inclusive : then it can be shewn that the number of df arbitrary constants involved in the solution is the greater of the two numbers m-\- s and n + r. See Cournot, Traite EU- mentaire de la Theorie des Fonctions... ISll. Vol. II. p. 318.] ^ ^ ^,. d^x , , d^ii , ,, Lx. 1. U-iven -^ = aaj 4- oy, -^^ = ax-\-by. 1st method. Differentiating the first equation twice with respect to f, we have d^x _ d'^x , d^y . , . d^y Eliminating y and -j4 from the above three equations, we have d^x d^x -^-(a+^>');^ ■]r{aV-ah)x=^0 {a). The complete integral of this linear equation with constant coefficients will determine a?, whence y is given by the formula 1 fd^x y=hKdf -''''. 2nd method. From the given equations we find d'^x d^y , ,, ,^ ,,, -^+w-^ = {a-\-ma)x-\-{h + ml>)y , ,. ( h + mh = {a + ma) [x -\ , \ a + ma ART. 10.] HIGHER THAN THE FIRST. 311 Let X 4- my — u, then provided that we determine m by the condition h + ml) m= -, (b). a + ma we shall have -~r^ = [a + ma ) u^ whence u = a^e^'^"^"'"^'^ + C^e-^-^'A Let m^, m^ be the values of w given bj (h), then the complete primitive system is X + m^y = (7^e^"^'«^"''^' + C^e-f""^'^'''^^', and this is really equivalent to the previous solution, though more symmetrical. Ex. 2. The approximate equations for the horizontal mo- tion of a pendulum when the influence of the earth's rotation is taken into account* are de ^"^ dt^ I -^ df dt I J («). I representing the length of the pendulum, g the force of gravity, and —r being equal to the product of the earth's angular velocity into the sine of the latitude of the place. As the equations have constant coefficients they admit of complete integration. If we diiferentiate so as to enable us to eliminate y, ~ and -~ , we find as the result dt' ^''v i) df ^r ^ ^' * Jullien, Prohlemes de Mecaniqiie Rafionnelle, Tom. n. p. 233. 312 EQUATIONS OF AN ORDER [CH. XIII. the complete solution of which is of the form x = Acos {ii\t + a) + B cos [m^t -f /3) (c) , where A, a, B, ^ are arbitrary constants, and m^^, m^ are the two roots, with signs changed, of the equation From the above value of x that of y may be obtained by means of the formula I d^x I ( ^ q\dx which is readily deduced from the given equations. The above system may also be solved by assuming x = x cos rt + y' sin rt ) + y' cos 7't) y — — x sm rt The transformed equations are (e). df ■+xv = o, 5' + xy = 0, where X^=.^+f; whence we find x — A cos \t-\- B sin \t -... 1 y = A' cos \t + B' sin X^ (/)• 11. In problems connected with central forces particular forms of the following system of equations present themselves, viz. d^_dR d^_dR d^_dR , V df ~"^' df ~ dy' df~dz ^^^' where i^ is a given function of the quantity ^/ [x^ + '^/ + ^^) ART. 11.] HIGHER THAN THE FIRST. * 313 or r. Multiplying the above equations by dx^ dy^ dz respec- tively, and integrating, we have H©"-©"-©]=^-^ «. B being an arbitrary constant. . . . dR dR dr x dR n . . Again, since -j~ = -^ -^ = ry , &c. the given system of equations may be expressed in the form d'^x _x dR d'^y _y dR d^^z z dR df ~ T dr' df ~V dr ' l^f ~ r dr ' Now if from each pair of equations we eliminate -,- , we obtain dSi d^x ^ d^^z d'^y „ d^x d^z ^ dt" ^ df ' ^ dt' dv ' dt' df ' of which it is evident that two only are independent. Inte- grating these, we have dy dx ^ dz dii .-, dx dz ^ dt ^ dt ^' "^ dt dt ^' dt dt -' Cj , Cg , C3 being constants. Squaring the last three equations and adding, we obtain a result which may be expressed in the form or, by virtue of (^) and of the known value of ?% dr 2r{R + B)-{^r^^j=A^ (c), rdr / rdr 314 EQUATIONS OF AN OEDER [CH. XIII. Again, it is evident that hy means of (c) we can eliminate B from each equation of the system {a). For (c) gives Substituting which in the first of the given equations, we have 'df~r\ T^ '^Tt dr dtj XT d'^x d\ A^x ^ Hence ^__^_+__ = 0, dt dt r T^ therefore r^ ^ r' 4 f-^ + ^' - = 0. dt dt \Tj T Adt If we now assume — ^ = d(^, the above becomes ^,fx r X whence - = a^ cos ^ + ^i sin ^ (/). In like manner, we find ^ = ^2 cos <^ + &2 sin (^ (^), ttgCOS (^ + Z^gSin^ .....(^), T Z r in which we must substitute for ^ its value, viz. CAdt _ f Adr .. ART. 12.] HIGHER THAN THE FIRST. 315 To this expression it would be superfluous to annex an arbi- trary constant before that substitution. For each of the second members of (/), [g), {h) is expressible in the form (7 cos ((/) + C), in which (j> is already provided with an arbi- trary constant The solution is therefore expressed by means of (e) and (/), which determine r and the auxiliary <^ as functions of t, and ^7 (/)j iSf)^ Wj '^hich then enable us to express x, y, z as functions of t. As we have however made no attempt to preserve independence in the series of results, the constants will not be independent. If we add the squares of (/), (^), (^), we shall have 1 = {a^ + a/ + al) cos* ^4-2 {ap^ + a]\^ + a^-^ sin <^ cos <\> which involves the relations among the constants The six constants in (/), (g), (h), thus limited supply the place of only three arbitrary constants, and there being three also involved in (e), the total number is six, as it ought to be. In the same way we may integrate the more general system d^x^ _ ^^ ^^^2 _ dR d^x^ _ dR ~W " dx^' It" ~dx„'"' df ~dx^' where i? is a function of V (x^^ -{-xj^ ..c +ir„^). The results, which have no application in our astronomy, are of the form which the above analysis would suggest. Binet, to wliom the method is due, has applied it to the problem of elliptic motion. (Liouville, Tom. Ii. p. 457.) For all practical ends the employment of polar co-ordinates, as explained in treatises on dynamics, is to be preferred. 12. The following example presents itself in a discussion by M. Liouville*, of a very interesting case of the problem of three bodies. * Sur un cas paiiiculier du Probleme des trois corps. Journal dc Mathcma- tiques, Tom. i. 2nd series, p. 248. 316 EXERCISES. [CH. XIII. Ex. Given -^ + w' {u - 3x {ux + vij)] = 0, Avliere, for brevity, x is put for cos [at+ h), y for sin (a^+ h). If we transform the above equation by assuming ux + vy — Uj uy — vx = F, we find, after all reductions are effected, And these equations being linear and with constant co- efficients, may be integrated by the process of the previ- ous section. EXERCISES. 2. J + 7a;-y = 0, J + 2x + 5y = 0. dx , t dy ^ „, 4, __ + oa; + 2/ = e« J + 3^-a, = e-. cZ^ J?/ , o. z , = ~ , = dt. '2y — 0X + e x — by + e CII. XTII.] EXERCISES. 6. 7 dy dz — dx= - — '- — = . 3?/ + 4^ 2ij + DZ 7. _|_3a,_4y + 3 = 0, ^,+x-Si/ + 5 = 0. 8. d^x d"z/ ^^ Sx 43/ + 3-O, -^+x + 7/ + o = 0. 9. 10 Given .^^ - ^y ^^ dt . . ^ . JL V/« ^''''^ X-^T," Y^Tr Z^T- T ^^^""'^ X= ax-\-'by ■{■ cz^ Y= ax + h'y + c'^, Z=a"a; + Z>"?/ + c"2r, and '. r, 2;, 7;, T3 are functions of t 11. What is the general form of the solution of a system of n simultaneous equations of the first order between ?i + 1 variables ? 12. What number of constants will be involved in tlie solution of a system of three simultaneous equations of the first, second and fourth order respectively between four variables ? 13. Of the system of dynamical equations, d'^x iix d^y iiy d^z iiz T^^+7^ = ^' ~df^-r'=^^ -df-^V^^^ where r = [x^ -^ y"^ + z'^Y, seven first integrals are obtained of which it is subsequently found that five only are independent. How many final integrals can hence be deduced without pro- ceeding to another integration ? )ISES. [CH. XIIT, a). c-l=ia-h)xy , \-^/j (3). 318 EXERC 14. Given a-j- =(b — c) yz l^f^^{c-a)zx (2), Puttinsr , = 1, =wi, r= n we find, on eliminating cfe, ^ 6-c c-a a-o ZxcZa; = fMjdy = nzdz, from which y and 3 wall be found in terms of x, and their values will reduce (1) to a differential equation of the first order between x and t. Or multiply the given equations, first by x, y, z, respectively, add the results and integrate ; 2ndly by ax, by, cz, respectively, add the results and in- tegrate. Then by means of the integrals obtained eliminate two of the varia- bles from any of the given equations. 15. Shew that in the example of Art. 12, the transform- ation x = x cos [rt -\- e) -\-y sin [rt + e), y = — x sin [rt -\- e) -{■ y cos {rt + e) , e being an arbitrary constant, would not lead to a more general solution than the one actually arrived at. ( 319 ) CHAPTER XIV. OF PARTIAL DIFFERENTIAL EQUATIONS. 1. Partial differential equations are distinguished by the fact that they involve partial differential coefficients in their expression, and therefore indicate the existence of more than one independent variable. Chap. i. Art. 2. The nature of these equations will be best explained by one or two examples of the mode of their formation. Ex. 1. The general equation of cylindrical surfaces is x — lz = (j>{y — mz) (1), and -v/r are arbitrary symbols of functionality. Proceeding to differential coefficients of the second order we find -£.2 = ^' W [y + «^) + V {y - «^)I» d'^z —2 = " [y + «^) + '^" {y - «^) , whence dy' d^ dx' = «^:^ (7), a partial differential equation of the second order and of the first degree. And. this equation has been formed by the elimination of two arbitrary functions from the general primitive. ART. 1.] TARTIAL DIFFERENTIAL EQUATIONS. 321 These examples illustrate the usual, and what may per- haps with propriety be termed the primary, modes of genesis of partial differential equations, viz. the elimination of arbi- trary functions, and the elimination of arbitrary constants. It is to be noted that these modes are perfectly distinct. Thus we might in Ex. 1, by specifying the form of the function , eliminate the constants I and m from the primi- tive (1), and the derived equations (2) and (3), instead of eliminating the functional forms from the two latter; but the result would differ in character, as well as in the mode of its origin, from that which has been actually obtained. We must bear in mind that when from a primitive equation of given form different partial differential equations are derived, it is owing to a difference of assumption as to what is to be re- garded as arbitrary; so that we are not permitted to say tliat to the same primitive, considered in the same sense of gencralitv, different partial differential equations belong. In Ex. 1, a partial differential equation of the first order has been formed from a general primitive containing one arbitrary function, and in Ex. 3 a partial differential equation of the second order has been formed from a general primitive contain- ing two arbitrary functions. These examples exhibit a certain analogy with the genesis of ordinary differential equations, the order of the equation being equal to the number of constants in its primitive. But this analogy is not general. For let be an assumed primitive containing two arbitrary functions {x) , It sometimes happens that equations not belonging to the above class are reducible to it by a transformation. Ex. 3. Given ^|- = x' + y\ Let -^ — w, then we have ax dw 2,2 whence integrating with respect to ?/, and adding an arbitrary function of ic, w = x'y-\-'J^-ir^{x). dz . Restoring to w its value ~ , integrating with respect to Xy and adding an arbitrary function of y, we have X y y^x ^^[x)dx-\-^^{y). Now ^ {x) being arbitrary, ^ ix) dx is also arbitrary, and may be represented by xi^)^ whence x^y + y^x ^ / \ . I / \ ^= .4 +X(^)+t(3/)- 21—2 324 LINEAR PARTIAL DIFFERENTIAL EQUATIONS [CH. XIV. Linear 2JCirtial differential equations of the first order. 3. When there are but three variables, z dependent, x and ?/ independent, the equations to be considered assume the form P, Q, and R being given functions of x, y, z, or constant. This form we shall first consider. Usually tlie differential coefficients ~ and -^ are repre- sented \>j p and q respectively. The equation thus becomes Fp-\-Qq=R (1). The mode of solution is due to Lagrange, and was first established by the following considerations. Since ^ is a function of x and ?/, we liave dz = 'pdx^qdy. Hence eliminating p between the above and the given equa- tion, we have Fdz - Edx = q{Pdy - Qdx) . Suppose in the first place that Pdz — Rdx is the exact diffe- rential of a function m, and Pdy — Qdx the exact differential of a function v, then we have du = qdv. Now the first member being an exact differential, the second must also be such. This requires that q should be a function of V, but does not limit the form of the function. Represent it by ^'(^)j ^^^sn we have du = (f)'{v)dv, whence ^ = *W (2). The functions u and v are determined by integrating the equations Fdz - Rdx = 0, JPdy - Qdx = 0, ART. 3.] OF THE FIRST ORDER. 325 sjmmetricallj expressible in the form dx _ dy _ dz ~P~~Q~ll ^^^' and of whicli the solution, Chap. xiii. Art. 5, assumes tlie form u = a, v = h (4), a and h being arbitrary constants. Dismissing the particular hypothesis above employed, La- grange then proves that if in any case we can obtain two integrals of the system (3) in the forms (4), then u = cf){v) will satisfy the partial differential equation, in perfect indepen- dence of the form of the function (j>. We shall adopt a somewhat different course. We shall first establish a general Hule for the formation of a partial differential equation whose primitive is of the form u = (p{v)y u and V being given functions of x, ?/, and z. Upon the solu- tion of this direct problem we shall ground the solution of the inverse problem of ascending from the partial differential equation to its primitive. Proposition. A primitive equation of the form u — (l>{y)^ where u and v are given functions of x, y, z, gives rise to a partial differential equation of the form Pp-\-Qq = R (5), where P, Q, R are functions of x, y, z. Before demonstrating this proposition we stop to observe that the form u = <^{y) is equivalent to the form fill, v) denoting an arbitrary function of u and v. For solving the latter equation we have u = (^{v). It is also equivalent to F[x,y,z,^{v)]=Qi, (j) being an arbitrary, but F a definite functional symbol. 326 LINEAR PARTIAL DIFFERENTIAL EQUATIONS [CH. XIV. For solving the latter equation with respect to {v)=F^{x, y, z), or {v)=u on representing F^{x, y, z) by u. Thus the proposition affirmed amounts to this, viz. that any equation between x^ y, and z which involves an arbitrary function will give rise to a linear partial differential equation of the first order. Differentiating the primitive w = ^ (t') , first with respect to X, secondly with respect to y, we have du du , , , , fdv dv du du ,, . . /dv dv Eliminating (f>' (v) by dividing the second equation by the first, we have du du dv dv dy dz _dy dz^ du du dv dv ' dx dz ■^ dx dz^ or, on clearing of fractions, (du dv du dv\ [du dv du dv\ \dy dz dz dy) ^ \dz dx dx dz)^ _ du dv du dv , . dx dy dy dx Now this is a partial differential equation of the form (5). For u and v being given functions of x, y and z, the coefficients of jt? and q, as well as the second member, are known. The proposition is therefore proved. As an illustration, we have in Ex. 1, Art. 1, u^x — lz, v—y — mz, whence ^,. J.. I, du du du dx~ = 1, dy^ = 0, dz dv dv dv dx = ^, dy = = i, dz -^ — — m. ART. 4.] OF THE FIRST ORDER. 327 Substituting these values in (6) there results, Ip ■\- mri = 1 , which agrees with the result before obtained. 4. The general equation (6), of which the above theorem is a direct consequence, has been established by the direct elimination of the arbitrary function. But the same result may also be established in the following manner, which has the advantage of shewing the real nature of the dependence of the coefficients P, Q, R upon the given functions u and ?;. Differentiating the equation u = (p{v) with respect to all the variables, we have du J du J dii ^ ,, , . fdv , dv y dv j and as this equation is to liold true independently of the form of the function ' (v), we must have du J du J du J „~1 dv T dv ^ dv J „ I ^ ^^ --dx + ^- dy-\- -r dz — 0 \ dx dy ^ dz J whence we find dx dy dz da dv du dv du dv du dv du dv du dv dy dz dz dy dz dx dx dz dx dy dy dx (9). Introducing now the condition that z is the dependent, X and y the independent variables, we have pdx 4- qdy — dz. To eliminate the differentials, let the terms of this equation be divided by the respectively equal members of (9), and we have 328 LINEAR PAETIAL DIFFERENTIAL EQUATIONS [CH. XIV. 'du dv du dv\ fdic dv du dv\ ^dy dz dz du) ^ \dz dx dx dzj " _ dii dv du dv . . ~ dx dy dy dx ^ '' ■which agrees with (6). Now if in the above general form we represent as before the coefficient of p by P, that of q by Q, and the second member by R, we see from (9) that P, Q, R are proportional to dx, dy and dz, in the system (8). But that system is precisely the same as we should obtain by differentiating the equations u = a, v = h, a and h being arbitrary constants. Hence, the partial differ- ential equation whose complete primitive is u = (j)(v), may be formed by the following simple rule. EuLE. Forming the equations u — a, v = &, where a and h are arbitrary constants, differentiate tJiem, and determine the ratios of dx, dy, dz in the form dx __ dy _dz . . 'F~'Q~R ^^^^* Then icill Fp+ Qq=R he the differential equation required. Or, tlie Eule may more briefly be stated thus. Eliminate dx, dy, dz leticeen the three equations, du = 0, dv=0, dz — pdx — qdy == 0 (12). It is worth wliile to notice that the partial differential equa- tion here presents itself, like many other results of analysis, in the form of a determinant. Ex. The functional equation of surfaces of revolution, the axis passing through the origin, is Ix -f- my + nz = lete determination of the factor rests. To complete the theory of the linear partial differential equation Pp + Qq = R it ought to be shewn that the solu- tion u =f(y), or as it may be expressed, F(u,v)=^0 (14), includes every possible solution. Let xi^j V) ^) = 0i 01* ^or simplicity % = 0, represent any particular solution. Differentiating, we have ^ + ^p = 0 ^ + ^^ = 0 dx dz-^ ^ dy dz^ ' and substituting the values of ^ and q hence derived in the given equation ax ay dz Similar equations being obtained from the particular in- tegrals u = a, v = b, we have, on eliminating P, Q, Ji, dx fdu dv ^du dv\ dx fdu dv du dv\ dx \dy dz dz 'dyj dy \dz dx~ dx dz) dz \dx dy dy dx) 332 LINEAR PARTIAL DIFFERENTIAL EQUATIONS [CH. XIV. Now suppose the forms of u and v to be u = ^[x,y, z), v = y\r{x,y, z) (16), ^{x, y, z) and -^{xy y, z) being given functions. From these two equations some two of the quantities x, y. z may be de- termined as functions of the other and of u and v. Suppose x and y thus determined as functions of z^ w, and v; then by substitution ^ (^j 3/j ^) becomes a function of z, u, and v, and we may write %(-^, J/>^)=Xl(^;«^;^)• Hence we find dx^djx^ du^dxy dv dx du dx dv dx ' dx ^dx^ d^ j^dx^ dv^ dy du dy dv dy ' dx ^do(^ du ^dx, dv^ _^dx, ^ dz du dz dv dz dz Substituting these in (15) and reducing, we have ^%i (^^^ ^^ du dv\ _ /. X dz \dx dy dy dx) But, were the second factor of the first member equal to 0, u would be a definite function of v and z (Chap. II. Art. 1) and the equations (16) could not determine x and y as by hypothesis they do. We have then —^ = 0, whence Xi does not involve z. Thus, X being expressible as a function of u and v, the equation ;!^ = 0 is included in the general form (14). 6. The above theory may be obviously extended to partial differential equations of the first order and degree involving any number of variables. ART. 6.] OF THE FIRST OIIDER. 333 Let cc^, x.y..x^ represent the independent A^rlables and z tlie dependent variable. Let moreover the primitive func- tional equation be expressed in the form u = (i>{i\, v^...n^;} (18), where u, v^, v,^...v^^ are known functions of the variables. Differentiating with respect to all the variables, and for brevity representing y deb y But (j) being an arbitrary function of tlie quantities i\, v^...i'„_^, it is evident tliat the supposition that the above equation is generally true involves the supposition that the system of equations du = 0, dvj^ = 0, di\^ = 0, . . .di\_^ = 0, is true, a system of which the developed form is du 7 du , du , -"I -^ - dx...,-\- — - dx„ + - 7 dz = 0 dx^ ^ dx^ dz dv. , dv, y dv, 7 ^ ^.^•••• + rf;^/^"+ J.<^"^^ (19). ^^^dx + '^'^> doc + ^ rf, = 0 Now this system may be converted into an equivalent sys- tem determining the ratios of the differentials dx^ , dx,,. . .dx^, dz, in the form '•'- '■' (20), dx^ _ dx^ _ where P^, P^...P^ and B are functions of the variabh^s or are constants. Introducing the condition that z is to be regarded as a func- tion of iTj, x^y...x,^, we have 2Vlx^+2\^X2...-^2\dx,, = dz (21), 334 LINEAR PARTIAL DIFFERENTIAL EQUATIONS [CH. XIV. wliere^,,^o.-.i?n are tlie several first differential coefficients of z. Ancl now eliminating the differentials dx^, dx^^...dx^, dz from (20) and (21) by division, we have Pdh + P.P."''rPnPn = I^ (22), for the partial differential equation sought. Conversely, to integrate the above equation it is- only neces- sary to form and to integrate the system (20). Representing the integrals of that system in the forms u = a, t\ = ^1, v^ = 0^, ... r^i = ^n-i 5 the final solution will be u = (f>i^i, V2'---0 ' (23)- This solution may also be put in the form ^{u,v„v,,...v^,) = 0 (24). y dt , . dt , . dt Ex. {y+z+t)-^-}-{z+x+t)--r+{x + 2j + t)-^^ = x + 9j-]-z, Lagrange, Memoires de VAcademie Royale de Berlin, 1779, p. i52. Here the auxiliary system of equations is dx _ dy _ dz __ dt 'y^\^Wt~ z^-x + t" x-\-y + t~ x + y + z' which is reducible to the form dt — dx _dt — dy _dt — dz _dx -V dy •\- dz -^ dt x-t "" y-t ~ z-t ~ 'd{x + y + z + t) ' each term being now an exact differential. The system of integrals will evidently be {x + y + z-\- t)^. x — t y — t z — t Or, representing the function x + y-{-z + t\ij S, sHx-t)=c,, sHi/'-t) = c„ sHz-t) = ART. 7.] OF THE FIRST ORDER. 335 Whence the complete integral symmetrically exhibited will be The solution of all partial differential equations of the form where Zj, X^^..,Xr^ and Z are any linear functions of the variables x^, x^,...x^, z, may be completely effected. For it depends on the solution of the system of ordinary differential equations dx^ _ dx^ _ dx^ _ dz which has been fully discussed in Chap. xiil. Hesse has integrated the still more general equation which, according to the above notation, would present itself in the form X — + X ^ + X ^ ^ dx^ ^ dx^ '" '* dx^ -.^ f dz dz ^^ \ __ V where X^, X2,...X^2 are any linear functions of the variables. (Crelle, Tom. xxv. p. 171.) Non-linear eqiiations of the first order with three variables, 7. Partial differential equations of the first order with two independent variables x^ ?/, and one dependent variable z, have for their typical form F{x,7/,z,p,q)=0 (1). Those which are linear with respect to p and q, we have considered apart. Those which are non-linear we proceed to 336 NOX-LIXEAPw EQUATIONS OF THE FIRST [CII. XIV. consider. The genesis of an equation of this class from a com- plete primitive involving two arbitrary constants has been illustrated in Ex. 2, Art. 1 ; and the mode is general. From a given primitive, involving a?, y, z with two arbitrary con- stants, and from its two derived equations of the first order formed by differentiating with respect to x and y respectively, it is possible to eliminate both the constants. The result is a partial differential equation of the first order. Conversely the integration of such an equation consists mainly in the discovery of its complete primitive — not that this is its only form of solution, but because out of it all other forms may be de- veloped. From the complete primitive involving arbitrary constants arise, 1st, the general primitive involving arbitrary functions; 2ndly, the singular solution. The terminology of Lagrange is here adopted. {Calcid des Fonctions, Lecon xx.) To deduce the complete primitive of a pcirtial differential equation of the form F{x, y, z, j). q) = 0. The existence of a primitive relation between x, y, z in- volves tlie supposition that the equation dz =pdx-\- qdy (2), sliould satisfy the condition of integrability, dy)~\dxl ^^'' wliere (^-] represents the differential coefficient of ^^ witii respect to y on the assumption that^:> is expressed as a func- tion of X and ?/, and f-^j the differential coefficient of ^ with respect to ^, on a similar assumption as to the expression of q. Xow regarding p for the sake of greater generality as a tunction oi x, y^ z, z being at the same time an unknown function of x and y, we have dp\ _ dp dp dz dy) dy dz dy dp dp ay dz AKT. 7.] ORDER WITH THREE VARIABLES. 337 Again, suppose that by means of the p:iven clifrerentlal equation, q may be expressed as a function of x, ?/, z, p. Re- garding in such expression z as a function of ^, y, and w as a function of ic, y, and z, we have idq\ _ dq dq dz dq fdp dp dz \dxl dx dz dx dp \dx dz dx^ dx dz ^ dp dx dp dz ^ ' Substituting these values in (3), we liave on transposition dp dx dy V dpi dz dx '^ dz ^ '' Now the coefficients — -, - , 1~]?'^ -> ^^^^ '^^'^ second member 7 +i^ / being known functions of x^ y, z, p, since q as determined by the given equation is such, the above presents itself as a linear partial differential equation of the first order in wliicli p is the dependent and x^ y, z the independent variables. Applying therefore Lagrange's process, Art. 6, we have the auxiliary system dx J dz __ dp dp ^ ^ dp) dx ^ dz and this, it is to be observed, is a system o{ ordinarji differen- tial equations between x, ?/, z^ and p. It may further be noted that while it has been formed in order to secure the integrability of the equation dz —pdx + qdy^ it also includes that equation. For it gives dz = [q - p '^ dy =pdx + qdy, since by the equation of tlie first and second members B. a E. 22 o. Substitute this value in the auxiliary system (5), and deduce hy integration a value of p involving an arbitrary constant. Substitute that value of p with the corresponding value of q in the equation dz=p)dx-{-qdy, also included in the auxiliary system (5), and again integrate. Ex, 1. Eequired a complete primitive of the equation z =pq. Substituting - for q^ the system (5) becomes z -^ 'Zz -^ The equation dp = dy gives p = y + a, whence q y-\-a Therefore dz — (u-\- a) dx + dy, of which the integral is z = [y + a)[x + h) (6), a and b being arbitrary constants. This then is a complete primitive. ART. 8.] ORDER WITH THREE VARIABLES. 339 Another will be found hy employing the equation integrating wliicli, we have p=cz~, ^ = -, whence dz = cz'i dx + — dy. c ^ Integrating, we find ^ A 1 2z- = cx-{--y-\-e, <5^' ^ = — — (0, r being a new arbitrary constant. It will be found on trial that both (6) and (7) satisfy the equation z —2^^- 8. Prop. Given a complete primitive of a partial differ- ential equation of the first order, to deduce the general primi- tive and the singular solution. Expressing the complete primitive in the form ^=/(«^5y. «^ ^) Wj a and h being its arbitrary constants, the partial differential equation is itself obtained by eliminating a and h between the above equation and the derived equations ^_df{x,y, g, h) _ df{x, y, a, h) ^~ dx ' ^' dy ' or, as we may for brevity write, if if m Now reasoning as in Chap, viii., the effect of the elimination will be the same if a and Z>, instead of being constants, are 22 2 340 KON-LINEAR EQUATIOXS OF THE FIEST [CIT. XIY. made functions of x and y^ so determined as to preserve to the equations (9) tlieir actual form. But a and h being made variable, we have elf df da df dh ^ dx da dx dh dx ' _df dfda dfdb_ ^ dy da dy db dy ' Hence the equations for determining a and h are d£da^df db^^ da dx db dx ^ ^' df da^df db^^ da dy db dy ^ ^' 'Now this system may be satisfied in two distinct ways, ] st by assuming !=«' f- (-)• The values of a and b hence found lead, on substitution in the complete primitive, to that solution which Lagrange terms singular. 7/* 7 /» 2ndly, Supposing --^ and -^ not to vanish, we have, on elimination of them from (10), (11), da db da dh _ ^ , ^. dx dy dy dx ^ Now this supposes either, 1st, that a and h are constant, which leads us back to the complete primitive; or, 2ndly, that h is an arbitrary function of a. Chap. ir. Art. 1. Again, multi- plying (10) by dx and (11) by dy, and adding, we have %da^-%dh = ^ (U). da db ART. 8.] OKDER WITH THREE VARIABLES. 341 rp riius the system (10), (11) is now replaced by tlie system (13), (U). Making then, in accordance with (13), h = ^(a), the expres- sion for z in (8) becomes while (14) becomes ^/[^, y, «, («)} = 0. And these together constitute what Lagrange terms the gene- ral jprimitive. To apply them it is only necessary to give a particular form to <^(a), and then eliminate a. Hence the fol- lowing theorem. Theorem. A complete primitive of a partial differential equation of the first order being expressed in the form ^=/{^>y;«, ^) (15), the general primitive loill he obtained by eliminating a between the equations ^^df{x,y,a,<^{a)\ (16), da J the singular solution^ by eliminating a and h between (15) and the equations df[x, y, «, ^) ^ Q df{x, y, a.'b) ^^ ^ ,^^x da ' db ^ ^' It will be observed that the process for obtaining the general primitive is virtually equivalent to that by which we should seek the envelope of the surfaces defined by the corresponding complete primitive, the constants a and b being treated as variable parameters connected by an arbitrary relation, while the ])rocess for obtaining the singular solution is that by whicli we should seek the envelope of (15), supposing a and b to be independent parameters. 342 KON-LINEAR EQUATIONS OF THE FIEST [CH. XIY. Tims, of the system of solutions which consists of a complete primitive, a general primitive, and a singular solution, the complete primitive must be regarded as forming the basis, and the system itself geometrically interpreted includes the surfaces represented by the complete primitive together with the whole of their possible envelopes. Ex. To deduce the general primitive and singular solution of the equation z =2^1- A complete primitive being z= {y + a) [x + l) {a), the corresponding general primitive will be expressed by the system z = {yJra)[x^(i>{a)] 1 0 = x-\-^[a)+{y-\-a)'[a)) ^ ^' from which a must be eliminated when the form of ^[a) is assigned. Another form of the complete primitive being [cx + ^ + ef ' = 1 W' the corresponding form of the general primitive will be W: from which c must be eliminated when the form of '^ (c) is assigned. To deduce the singular solution, we have from (a), — = a* + & = 0, da dz ART. 9.] ORDER WITH THREE VARIABLES. 343 Hence, h = — x, a = — y which, substituted in (a), gives z = 0, a singular solution. The same result is deducible from (c). 9. In the last example, two complete primitives, two cor- responding forms of general primitive, and one common form of singular solution are presented. Two systems of solution appear, and the question arises: Does either system suflicc alone .^ The answer is given in the following theorem. Theorem. All possible solutions of a i:>artial differential equation of the first order, are virtually contained in the system consisting of a single complete primitive, with the derived gene- ral ptrimitive and singular solution. As before, we shall represent the proposed differential equa- tion and its given complete primitive in the forms, F{x,y,z,p,q) = {) (18), z=f{x,y,a,h) (19). We shall also represent in the form, ^ = %(^.^) (20), some solution of (18), of which nothing more is known than that it is a solution. We are to shew that such solution is included in the system of solutions of wliich the common primitive (19) constitutes the basis. If we represent for brevity the values of 2; in (19) and (20) by /' and ^ respectively, we shall have, since both are solu- tions of (18), ^[-'y'f''i'f)=' c^^)' ^(-^'<4'|)- (-)• From the form of the above equations it appears tliat if a and h are so determined as to satisfy two of the conditions. (^f _ <^h ^f ^.Y -^ '^' dx dx ' dy ay ^ 9?.^ 344 K OX-LINEAR EQUATIONS OF THE FIRST [CH. XIV. they will satisfy the third. For suppose they satisfy the first two, then the system (21), (22) may be expressed in the form in which the truth of the third equation of (23) is involved. Now, as (19) satisfies (18) whatever constant values we assign to a and Z>, it still will do so if, after the differentiations 7 /• 7/» by which y- and — are found, we substitute for a and h any functions of x and y. But a and h can be determined so as to satisfy two con- ditions. Hence they can be determined so as to satisfy the system (23). Differentiating the equation /=% on the hypo- ilicsis that a and h are functions so determined, we have df df da df dh _ d^ dx da dx db dx dx^ df dfda df_df^_d^ dy da dy dh dy dy ' 7/» 7 /> Here, -j- , -y- have the same values as in (23), being ob- tained by differentiating as if a and h were constant. Hence, reducing by (23), we have df da df dh _ ^ da dx db dx df da df db _^ da dy dh dy (25). But these are the equations (10) (11), Art. 8, by which the system of solutions founded upon the complete primitive is constructed. The argument then is briefly this, li z = x (^' V) ^^ ^ solution of the given partial differential equation, it is possible to determine a and b in the given complete primitive so as to satisfy the equations (23) ; therefore so as to satisfy the ART. 9.] ORDER AYITH THREE VARIABLES. 345 equations (25) ; therefore so as to indicate a necessary in- clusion of z~x (^5 y) ^^^ ^^^^ system wliicli is founded upon the given complete primitive. CoK. 1. Hence the connexion of a given solution with a given complete primitive may be determined in the following manner. Adopting the foregoing notation, determine the values of a and h which satisfy the system (23). If those values are constant, the solution is a particular case of the complete primitive ; if they are variable, but so that the one is a function of the other, the solution is a particular case of the general primitive ; if they are variable and unconnected it is a singular solution. Cor. 2. Hence also any two systems of solutions founded upon distinct complete primitives are equivalent. For each is virtually composed of all possible particular solutions. Ex. The equation z =^2', has for its complete primitive z — {x-\-a) (^ + ^)j and for a particular solution z — ^^^^——^ . What is the connexion of this solution with the complete primitive ? We have by (23), (^ + «)(y + ^)= J — , y-\-x y + x These equations are not independent, the first being the product of the last two. Any two of them give a = = y^^ I = ^~y whence 5 = — a. Thus, the values of a and h being variable, but such that J is a function of a, the proposed solution is a particular case of the general primitive. Some general questions, but of minor importance, rclatinG: to the functional connexion of different forms of solution, will be noticed in the Exercises at the end of this Chapter. 346 DEEIVATION OF THE SINGULAE SOLUTION [CH. XI Y. In quitting this part of the subject, we may observe that there are two modes in which the questions it involves may be considered. The first consists in shewing that the gain of generality, which in Charpit's process accrues in the trans- ition from the complete to the general primitive, is equal to that which Lagrange's original but far more difficult process secures by the employment of the general value of ^j> drawn from (4), instead of a particular value drawn from its auxiliary system. The proof of this equivalence, as developed with more or less of completeness, by Lagrange and Poisson {Lacroix, Tom. II. p. 564, III. p. 705), and recently by Prof. De Morgan {Cambridge Journal, Vol. vil. p. 28), is, from its complexity, unsuitable to an elementary work. The other mode is that developed in the foregoing sections. Derivation of the singular solution from the differential equation. 10. The complete primitive expresses z in terms of x, ?/, a, h. The differential equation expresses z in terms of x, y, p, q^. Either is convertible into the other by means of the two equations derived from the complete primitive by differ- entiating with respect to x and y respectively. Hence it is not difficult to establish the two following equations, dp dz dg dz da d'z _ dhdy dz d db Jo ^'z idy d'z d'z d'z d'z dadx dbdy dz d'z da dhd. dady dz V db dhdx d'z dadx d ''z d'^z d' z d'z dadx dhdy dady dhdx ;26), in the first members of which z is supposed to be expressed in terms of x, y, p, q by means of the differential equation, in the second members, in terms of x^ ?/, a, h by means of the complete primitive. ART. 11.] FROM THE DIFFERENTIAL EQUATION. 347 Now the singular solution is deduced from the complete primitive by means of the ecjuations 5« = ^' dh = '' ^')' and it is evident from the form of (26), that this will generally involve the conditions 1 = 0' I- (-)• Such then will generally be the conditions for determining the singular solution from the differential equation. The conditions (28) will not present themselves, should the denominator of the right-hand members of (26) vanish identi- cally. But it may be shewn that in this case the conditions (27) do not lead to a singular solution. And analogy renders it probable that icJienever the conditions (28) are satisfied the result, if it be a solution at all, will be a singular solution. The complete investigation of this point, however, would in- volve inquiries similar to those of Chapter Yiii. The Kule indicated is then to eliminate p and q fi-om the differential equation hy means of the equations (28) thence de- rived. 11. The following geometrical applications are intended to illustrate the preceding sections. Ex. 1. Eequired to determine the general equation of the family of surfaces in which the length of that portion of the normal which is intercepted between the surface and the plane ic, ?/, is constant and equal to unity. As the length of the intercept above described in any sur- face is z (1 +^/ + q") -> we have to solve the equation z'{l-\-f + q') = l {a). Hence q = {z^- 1 —p^)'-, and the auxiliary system (5), Art. 7, becomes, on substitution and division by (^~^— 1 —V^'-) dx _ dij _ ^^ _ ^^^^ n \ 348 DERIVATION OP THE SINGULAR SOLUTION [CH. XIV. From the last two members we have on integration c (1 - z')- ^-—z • Substituting this, with the corresponding value of q^ derived from (a)j in the equation dz =])dx-\- qdy wx have dz= -^ — -^ — + (1 - cy — ~-^y^ s z integrating which in the usual way, we find (1 _ ^2)i = _ c^ _ (1 _ ^^fy _ c', or, changing the signs of c and c', {\-z^f = cx-i,\-efij-\-c (c), which is a complete primitive. The corresponding form of the general primitive will be (l_,^)^=ea._(l_e^)^^ + ^(e) 1 0 = ic + c{l-c')~2^ + ^'(c)i from which c must be eliminated. ..-*, i w. But another system of solutions exists ; for from the first, third, and fourth members of (6) we may deduce jpdz + zd]^ + c?^ = 0, whence p^ -f a; = a, from which, and from the given equation determining^ and ^, we have to integrate dz = dx + ^ — dy. z z '^ The result is (^-ar+(y-jr+^^=i (.), a complete primitive. The corresponding general primitive is ART. 11.] FROM THE DIFFERENTIAL EQUATION. 349 To deduce the singular solution from the differential equa- tion {a) we have I = -^ (1 +/ + 2=) -f = 0, I = _ 2 (1 +/ + r/P = 0, whence p = 0, 2' = 0 ; substituting which in (a) we find Tlie above example illustrates the importance of obtaining, if possible, a choice of forms of the complete primitives. The second, of those above obtained, leads to the more interpret- able results. It represents a sphere whose radius is unity and whose centre is in the plane x, ?/, while the derived general primitive represents the tubular surface generated by that sphere moving but not ceasing to obey the same conditions. The singular solution represents the two planes between wliicli tlie motion would be confined. A\\ these surfaces evidently satisfy the conditions of the problem. Ex. 2. Required to determine a system of surfaces such that the area of any portion shall be in a constant ratio {in : 1) to the area of its projection on the plane xy. The differential equation is evidently 1 +;/+ (f = m^, and it will readily be found that it has only one complete primitive, viz. z = ax + \/{m^ -a^ — l)y + h. Thus the general primitive is z = ax + sjiiii" - a^ - 1) 7/ + (f) (a), and this represents various systems of cones and other develop- able surfaces. 350 sym:metiiical and general solution [ch. xiy. Similar but more interesting applications may be drawn from the problem of the determination of equally attracting surfaces. 12. Attention has already been directed to the different forms in which the solution of a non-linear equation may sometimes be presented. It may be added that linear equa- tions admit generally of a duplex form of solution. The ordi- nary method gives directly the equation of the system of surfaces which they represent; Charpit's method leads to a form of solution which exhibits rather the mode of their genesis. Ex. Lagrange's method presents the solution of the equa- tion {mz — ni/) p + {nx — ^2) q = Jy — mx (a), in the form lx-\-my -^-nz = ^ {x" + y^ + z"^) (h), the known equation of surfaces of revolution whose axes pass through the origin of co-ordinates. Charpit's method presents as the complete primitive of {a) {x — cVf-{- (y — cmy+ {z —cnY = r'^ (c), c and r being arbitrary constants. This is the equation of the generating sphere. The general primitive represents its system of possible envelopes. These solutions are manifestly equivalent. Symmetrical and more general solution of partial differential equations of the first order, 13. The method of Charpit labours under two defects, 1st, It supposes that from the given equation q can be ex- pressed as a function of a?, y] z, p] 2ndly, It throws little light of analogy on the solution of equations involving more than two independent variables — a subject of fundamental import- ance in connexion with the highest class of researches on Theoretical Dynamics. We propose to supply these defects. ART. 13.] OF PARTIAL DIFFERENTIAL EQUATIONS. eJ51 It will liave been noted tliat Cliarpit's method consists in determining ^9 and ^ as functions of x, ?/, z, which render the equation dz ■=j)dx + (idy integrable. This determination pre- supposes the existence of two algebraic equations between x^ y, z,p, q-, viz. 1st, the equation given, 2ndly, an equation obtained by integration and involving an arbitrary constant. Let us represent these equations by F{x,y, z,p, q) = 0, ^{x,y, z,]j, q)=a.... (20), respectively. And let us now endeavour to obtain in a general manner the relation between the functions F and ^. Simply differentiating with respect to x, ?/, z, p, q, and re- dF d<^ dF d^ presenting ^-by X, ^ by X\ ^^;by P, ^ by P', &c. we have Xdx + Ydy + Zdz + Pdp + Qdq = 0, X'dx + Ydy + Z'dz + Pdp + Qdq = 0 ; or, substituting j^fZa; + qdy for dz, {X+jyZ) dx + ( r+ qZ) dy + Pdp + Qdq = 0... (30), {X'+pZ')dx+{Y' + qZ')dy+P'dp+ Q'dq = 0...{Zl). But, representing for brevity ^^^ , ^^-|- and ^ , by r, 5, t, respectively, we have dp = rdx + sdy\ dq = sdx + tdy ) .' ^ Substituting these values in (31) we have {X'+pZ^rF + sQ')dx + {Y' + qZ' + sP'+fQ')dy = 0, which, since dx and dy are independent, can only be satisfied by separately equating to 0 their coefficients. These furnish then the two equations -{Y' + qZ')=sP + tQ'] ^'^'- 352 SYMMETRICAL AND GENERAL SOLUTION [CIT. XIY. IN'ow these equations are of tlie same /or??i as (32). Tliej establisli the same relations between the functions -{X'+pZ-), -[Y' + aZ'), P, Q, (34), as (32) does between the differentials dp, dq, dx, dy. It follows that if we give to dx and dy, wdiich are arbitrarj, tlie ratio of the last two of the functions (34) then will dp and dq have the ratio of the first two, so that the following w^ill be a consistent scheme of relations, viz. dx_dy_ dp _ dq P'- Q ~ X'-^pZ' - Y' + qZ' ^^''^- Now dividing the successive terms of (30) by the successive members of (35) we have {x+pZ) p' + ( r+ qZ) Q'-p {X' +pZ') ~Q{Y'-i-qZ')=0 (36). This is the relation sought. It might be obtained by direct elimination by multiplying the equations of (33) by P and Q respectively, and the corresponding equations derived from (30) by P' and Q' respectively, and subtracting the sum of the former from the sum of the latter. It is obvious too, and the remark is Important, that we might pass directly from (30) to (36) by substituting for dx, dy, dp, dq, the functions of (3-4), and that this substitution is justified by the identity of relations established in (32) and (33). If in (36) we substitute for X, Y, &c. their values, and transpose the second and third terms, we have '^ ^\^_(^ ^\ ^ {'^^ ^\ ^ ^dx -^ dz J dp \dx ^ dzj dp \dy ^ dz J dq d^ d^\ dF ^ -dy^^^z)^q=' (^")- Such is the relation which connects the functions F and ^. When F is given it assumes the form of a linear partial differ- ART. 14.] OF PARTIAL DIFFERENTIAL EQUATIONS. 353 ential equation of the first order for determining ^. If from its auxiliary system we can deduce any integral involving an arbitrary constant, and such that in conjunction with the given equation it enables us to determine /> and q as functions of X, y, z, the subsequent integration of dz —j[)dx + qdy will lead to a form of the complete primitive. 14. Analogy now points out the method to be pursued for the solution of equations involving more than two inde- pendent variables. Prop. To deduce the complete primitive of the partial differential equation F {x^, x.^...x^, z, iJ^, jp^...^^;) ^^ (38), , dz dz where ;,, = — , ...^„=„ . In the first place we must seek to determine values of ^7j, 2?2? .-.pn ill terms of the primitive variables x^, x^...Xn, z, such as will render integrable the equation dz =Pidx^ -\-^^dx^... -i-j^dx,, (39). Suppose one of the equations requisite in conjunction with (38) for this determination to be ^ (x^, x^, ...x^, z, 2\,p,, ...2h)=a, (iO). Then representing the first members of (38) and (40) by their characteristics F and ^, difi:erentiating, and substituting for dz its value given in (39), we have results which may be thus expressed, ^ [fdF dF\ , dF , ] ^ H[d^^^^Tz)^^^+d^M=' ^''^^ ^ (fd^ d\ , d^ , ] ^ H[d^^p^-d^)'^^'^dpM-'> where Sj represents summation from z = 1 to i = }i. J)ut smce^i = — , we have B.D.E. 23 354 SYMMETRICAL AND GENERAL SOLUTION [CH. XIV. ^■^^^~ dx^dx^ ^ dx^dx,^ ^"' dxidx,, »•••••• V Substituting this value in (42), we shall be permitted, in con- sequence of the independence of the differentials dx^, dx^,...dxn, to equate their respective coefficients to 0. It is easy to see that the coefficient of dx,. will be d^ d^ ^ d^ d'z dxj. '-^ dz ' dpi dx^dXr ' Equating this to 0, we have, on transposition, __ / J^ d^]_^ d'^z d^ \dXj. -^ dzj ^ dxidxj. dpi ' Hence, changing i into r and r into t, -(^+ .^) = t-^^— (44) \dxi ^* dz ) ^ dXfdxi d])^ Now comparing this with (43), and observing that d'z ^ d'z dxidx^ dxydxi ' we see that the systems of differentials represented by dp^ and dxy. respectively are connected by the same relations as the systems of functions represented by (d^ , d^\ . d^ ^. , Hence, by the reasoning of the previous example, it is per- mitted to substitute in (41), for the differentials, the correspond- ing functions, viz. — f-j— + 2^i -17 ) for dpi\ and -7— for dxi. We thus find H[d^,-'p^i^)df-dfXd^,^'''dz)r' (^^)' ART. 14.] OF PARTIAL DIFFERENTIAL EQUATIONS. 355 the summation extending from t = 1 to ^ = n. This is the relation sought, and it is seen to be symmetrical with respect to i^and <^. AVlien F is given, it becomes a linear partial differential equation for determining ^. From its auxiliary system of ordinary differential equations it suffices to obtain n— I integrals, ^i = «x. ^•a=«2. •••^«_i = ««-i (-4G), such as, in conjunction with the given equation, will enable us to determine ^^, ^a^, ...j?„ in terms of the original variables; then integrating (39), we sliall obtain the complete primitive in the form f{x^, a:,,...a?„, s, a„ o.,, ...oj =0.... (47). All other forms of solution are hence deducible by regarding a^, a^, ... a„ as parameters varying, independently or in sub- jection to connecting relations, but so as to leave unaltected tliQ forms of j;^ p.,, .../?„. It is proper to observe that the given equation F= 0 is itself included among the particular integrals of (45). In fact F is one of the forms of 4> which make 4> = a a solution, as will be found on trial. The given equation is therefore a particular integral. And therefore the n — l integrals of the system (46) must be independent of it in order to render the determination of^;^, 2\,, •••i\ possible. The equation (45) may be expressed as follows : ^ (dFd^ dFd^\ dF^ d^ d^^ dF_ ^' [dx, dp, dp] dxj "^ dz ^'^' dp, " dz ^'-^>, ~ ^- And under this elegant form, obtained however by a more complex analysis, the solution is presented by Brioschi [Tor- tolini, Tom. vi. p. 426, Intorno ad una ])ropr{eta delle cqua- zioni alle derivate parziali del primo or dine). The problem of the integration of partial differential equa- tions of the first order, irrespectively of the number of the variables, appears to have been first solved by Pfatf, but the 23—2 356 SYxMMETlUCAL AND GENERAL SOLUTION [CH. XIV. most complete discussion of it will be fomid in a memoir by Caucliy [Exercices d' Analyse, Tom. II. p. 238. Sur Vintegra- tion des equations aux derivees ijartielles dii ^premier ordre), in which the determination of the arbitrary functions of the general primitive so as to satisfy given initial conditions is fully considered. The connexion of the subject with Theo- retical Dynamics was first established by the researches of Sir W. Hamilton and Jacobi. The truth, illustrated above, that the solution of a partial differential equation of the first order is reducible to that of a system of ordinary differential equations, and the truth that the solutions of certain systems of differential equations (including that of dynamics) may be reduced to the discovery of a single function defined by a partial differential equation, are correlative. The researches above referred to, together witli those of Liouville, Bertrand, and Bour, founded partly upon their results and partly upon the allied discoveries of Lagrange and Poisson concerning the variation of the arbitrary constants in dynamical problems, contain the most important of recent additions to our specu- lative knowledge of Differential Equations. For this reason we have dwelt upon their history. Fuller information will be found in Mr Cayley's excellent Beport on the recent Pro- gress of Theoretical Dynamics. {Rejport of British Associa- tion, 1857.) [In an Appendix to the first edition Professor Boole pre- sented Art. 14 in the following form.] Art. 14. The most important form of the problem of this Article is the following, and the reader is requested to substi- tute it for the one in the text, sufficient account not being there taken of the conditions among the constants. Required a value of z as a function of x^, x^, ... x^ which shall satisfy the partial differential equation F[x^, x^,...x^, z,i\,ix^, ...^„)=0 --.(I), and shall, when ic„=0, assume 2. given form, z^^{x^,x^,...x^_:^.. ...., (2). Ar.T. M.] OF TAKTIAL DIFFERKXTIAL EQUATIONS. 3.j7 Eepresenting the second member of (2) by , and '^ by ^^5 &c., we shall have, when x,,= ()^ i\=i\^ i\='i>.^ .••7>«-i = «.!• (3), for, in seekinjr the forms wliich -r^. -7— ,.•• -7—-- assume when ax^ dx^ (lXn_^ x^= 0, we are permitted to make x^= 0 in the general value of z before differentiating. Kow the auxiliary system of the linear equation, (45) in the text, yields 2n integrals connecting ic^, ... a?„, 2, ^^^ .../>„ with 271 arbitrary constants. But since one of the integrals is i^=c, and since to make this agree- with (1) we must have c = 0, the 2n integrals will effectively contain 2?i — 1 arbitrary constants. This however being the number of the variables contained in (2), (3), namely of the variables a?i, ... a7„_i, ^, ^j, ... 2?„_j, we may express, and so replace, these arbitrary constants by initial values of the above vari- ables corresponding to cc„= 0. Let fi,.--?n-i? ?5 T^'ij-'-TTrt.! be the new constants in question; then, substituting these for the variables whose initial values they represent in the n equations (2), (3), we obtain n condi- tions connecting the above constants. Thus we have finally 3?i equations, consisting of 2n inte- grals with n equations of condition connecting the 2?i— 1 constants which those integrals contain. From these 3n equations we can eliminate the above In — 1 constants toge- ther with the n quantities p^^ p^,... p^- '^^^^ result will be a final relation between z, x^, x^,...Xn, which will be the solu- tion sought. If we regard the function 4> {x^, a-g' ••• ^"-1) ''^^ arbitrary, the above solution will constitute a general primitive ; but if we give to it a particular form involving n arbitrary constants, we shall obtain a complete primitive. (Gauchy, Exercices, Vol. II. p. 238.) )58 EXEECrSES. [CII. XIV EXEECISES. 1. How are equations, in whicli all tlie differential coeffi- cients have reference to only one of the variables, solved ? dz V 3. dx ^f-ar)' dz _ y dx x-i- z' 4. The ])artial differential equation of the first order whicli results from a primitive of the form u=f{v), where ii and v are determinate functions of x, t/, z, is necessarily linear. Prove this. 5. Ojp-\-hq = l. 7. 7JJ) + Xq=Z. 8. x^p — X2/q -\-y^ = 0. 9. Integrate the equation of conical surfaces {a — x)p-\- ip —y) q= c — z. 10. xzp + yzq = xy. 11. (/+ z"' - ar)p - 2xyq + 2xz = 0. 12. Required the equation of the surface which cuts at right angles all the spheres which pass through the origin of coordinates and have their centres in the axis of x. It will be found that this leads to the partial differential equation of the last problem. ^f 13. z —xy —yq — a {x^ + 3/^ + z^) ^, CH. XIV.] EXERCISES. 359 14. Find the equation of tlie surface wliIcK cuts at right angles the system of ellipsoids represented by the equation where D is the variable parameter. Lacroix, Tom. ii. p. C78. 15. Find the equation of a surface which belongs at once to surfaces of revolution defined by the equation ijy — qx = {), and to conical surfaces defined by the equation j)x-\-qij = z. In problems like the above we must regard the equations as simultaneous, determine jp and 5 as functions of x, y, z, and substitute their values in the equation dz=pclx + qdy, which will become integrable by a single equation if the problem is a possible one, but not otherwise. ^- ch dz dz xif 17. Explain the distinction between a complete primitive and a general primitive of a partial differential equation of the first order. 18. Find the complete and the general primitive of z =j)X + qij -\-pq. 19. Deduce a singular solution of the above. 20. i^:?=l. 21. q = x2o-{-f, 22. ShcAV from the form of its integral that {a) is given, the elimination of a from both will give a differential equation for determining the form of i/r (c) . 26. The equation z =2)q has two general primitives, 1st. z = {y + a)[x+cj,{a)], 0 = —[{y + a] [x + cj, {a)}], 2nd. 4.z = [cx + ^+'ylr{c)Y, 0 = -^ {ex + ^ + f{c)Y; shew hence that the relation between {a) and yjr (c) is ex- pressed by the equations cj>'{a) + \ = 0, cylr{c)-c'f'{c)=2a. ( 2G1 ) CHAPTER XY. PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 1. The general form of a partial differential equation of the second order is F{x,y, z,p, q, r, s, t)=0 0), dz dz d'z d'z d'z where p = -r •> <1= -r ^ ^' = 7~2 ? ^ = ~i — t" 5 ^ = t^i • ^ dx ^ dy dx'^ dxdy' dy' It is only in particular cases that the equation admits of in- tegration, and the most important is that in which the differ- ential coefficients of the second order present themselves only in the first degree ; the equation thus assuming the form Rr^^s^Tt=Y. (2), in which i?, >S', T and V are functions of x, y, z^ ]) and q. This equation we propose to consider. The most usual method of solution, due to Monge, consists in a certain procedure for discovering either one or two first integrals of the form »=/W (3), u and V being determinate functions of x^ ?y, -s", p^ q, and f an arbitrary functional symbol. From these first integrals, singly or in combination, the second integral involving two arbitrary functions is obtained by a subsequent integration. An important remark must here be made. Mongc's metliod involves the assumption that the equation (2) admits of a first integral of the form (3). Now tliis is not always the case. There exist primitive equations, involving two arbitrary func- tions, from which by proceeding to a second differentiation both functions may be eliminated and an equation of tlie form (2) obtained, but from wliich it is impossible to eliminate 362 PARTIAL DIFFERENTIAL EQUATIONS [CH. XV. one function only so as to lead to an intermediate equation of the form (3). Especially this happens if the primitive involve an arbitrary function and its derived function together. Thus the primitive leads to the partial differential equation of the second order r-t = 'i (5), but not through an intermediate equation of the form (3). It is necessary therefore not only to explain Monge's method, but also to give some account of methods to be adopted when it fails. 2. It is not only not true that the equation (2) has neces- sarily a first integral of the form (3), but neither is the converse proposition true. We propose tlierefore, 1st, to inquire under what conditions an equation of the first order of the form (3) does lead to an equation of the second order of the form (2) ; 2ndly, to establish upon the results of this direct inquiry the inverse method of solution. And this procedure, though some- what longer than that usually followed, is more simple, because exact and thorough. Prop. 1. A 2^cirtial differential equation of the first order of the form u=:f{v) can only lead to a partial differential- equation of the second order of the form Br + Ss + Tt=V. (6), when u and v are so related as to satisfy identically the con- dition du dv du dv . . dp dq dq dp For, differentiating the equation xi —f{v) with respect to a;, and observing that -j-=lo, 7/ ~ *'' i ~ ^> ^^^® ^^^^Q AET. 3.] OF THE SECOND ORDER. SG3 du dit du du ^, , , (dv dv dv dv\ ^ ^-^'cfo ^'"^ + ^ ^ =-^ ('')U +^^fe +''^7^ + ^V In like manner differentiating ic=f(v) Avitli respect to y, we have da du du , '^^ _ /•'/ n A^^ dv dv f^^\ dij -'■ dz dp dq '^ ^ ' \(2y -^ dz dp dqj ' Eliminating /' (v) there results fdu du du du\ /dv dv dv dv\ \ clx ^ dz dj? dq) \dy ^ dz dp jdqj dv dv dv dv\fdu du du du\ ^ ,,^. On reduction it will be found that the only terms involving r, s, and t in a degree higher tlian the first will be those which contain r^ and s^ The equation will in fact assume the form Br + Ss+Tt+ U{rt-s')=V (9), in which U=-j- ^ — ^. The forms of the other co- aj) dq dq dp efficients it is unnecessary to examine. Now tliis equation assumes the form (G) when the condition (7) is satisfied — and then only. 3. The proposition might also be proved in the following manner. Since u =f{v) we have du =f' (v) dv, an equation which, since f(v) is arbitrary, involves the two equations du = 0, dv = 0. Hence du -, du , die -, du -, du , ^ "1 -^~dx-\- -j- dv -\- -J- dz -\- -T- dp + -J- dq = 0 dx du ^ dz dp -^ dq -^ , ^ -^ ^ ^ [-...(10). dv -, dv y dv ^ dv J dv ^ 364 PARTIAL DIFFERENTIAL EQUATIONS [CII. XV. Bat dz =pdx + qdy^ dp = rdx + sdy^ dq^ = sdx + tdy. Whence on substitution [dii du du dii\ , fdn du dii du\ , dv dv dv dv\ , /r/?j dv dv dv\ , ax ^ dz dp dqj \dy ^ dz dp dqj "^ Whence eliminating dx and dy, we have the same result as before. 4. A consequence, which, though not affecting tlie present inquiry, is important, may here be noted. It is that it would be in vain to seek a first integral of the form u —f (v) for any partial differential equation of the second order which is not of the form (9). Prop. 2. To deduce when possible a first integral, of tlie form 2i=f(v), for the partial differential equation (6). By the last proposition u and v must satisfy the condition (7), which is expressible in the form, du _ du dv dv . aq ' dp dq ' dp Hence, if we represent each member of this equation by m, we have du _ du dv dv , . dq dp'' dq dp Substituting these values in (10), we have du ^ du ^ du ^ du , -, , . J" (13)* dv J dv J dv J dv ,j , ,. .[ '' and we are to remember that this system, being equivalent to du = 0, dv — 0 modified by the condition (7), can only have an integral system of the form, u = a, v = h (1-4), AllT. 4.] OF THE SECOND OKDER. 'dG5 a and b being arbitrary constants, and u and v connected by the condition (11). Making dz = j)dx ■\- qdij in (13), we liave d)i diC\ 7 (du diC\ , du , ^ -, ^ ^ dx ^ dzj ' \d]j ^ dzj '^ dj) I /,.v fdv dv\ 7 . fdv dv\ y dv , ^ , , ( U +^ rfi j '^^ + (^ + !? cfo j '^^ + d^ ^'^P + '"^^) = "^ ' From these and from tlie equations dp = rdx -{■ sdij , dc[ = sdx + tdy (IG), if we eliminate the differentials dx, dij, dp, dq, we sliall necessarily obtain a result of the form (6). For in thus doing we only repeat the process of Art. 3, with the added condition (7). To effect this elimination, we have from (16), dp -\- mdq = (r + ms) dx + {s -{- mt) dij ; or, rdx + s [dy + mdx) + tnidy = dp-\-md(j[ (17). Now tlie system (15) enables us to determine the ratios of dy and dp-\-mdq to dx, and these ratios substituted in (17), reduce it to the form (G). But in order that it may be, not only of the form (6), but actually equivalent to (G), it is necessary and sufficient that we have dx _ dy 4- mdx _ ondy _dp + mdq . H~^~S ~~W'~ V ^^^^• This system of relations among the differentials must thus include the equations (15). The same system (18), together with the equation dz =pdx + qdy, must therefore include the system (13). It must therefore in its final integral system include the equations u = a, v = h with their implied con- dition. 'SQQ PARTIAL DIFFERENTIAL EQUATIONS [CH. XV. We conclude then, that if the equation Rr ■\- Ss -\- Tt = V, result from an equation of the first order of the form u=f{v), the system (18), together with the equation, dz —jpdx + qdy (19), must admit of an integral system determining u and v in equations of the form u = a, v = h. To eliminate m from (18) we have, on determining its value from the first and third members, substituting it in the second and fourth, and reducing, Fidif-8dxdy+Tdx^ = 0 ...(20), Rdpdy+Tdqdx- Vdxdy = 0 (21), and these, with (19), make three ordinary differential equations among the five variables x, y, z, p^ q. But among five vari- ables there ought to exist four ordinary differential equations in order to render the final relations determinate. And this confirms what was said in Art. 1, of the hypothetical character of Monge's method. It is only when the proposed equation originates in an equation of the form u=f(v), that the above system admits of two integrals of the form, u = a, V = h. As (20) is of the second degree it will, unless it is a com- plete square, be resolvable into two equations of the first degree, and eitlier of these in conjunction with (21) and (19) may lead to a final integral system determining u and v. It follows that wlien the given equation admits of a first integral at all, it will admit of two such — excepting the case in which (20) is a complete square. 5. As yet no account has been taken of the quantity m. The mode in which it is involved in the equation (18), leads however to a remarkable consequence developed in the follow- ing Proposition. Prop. If by the last proposition we obtain two first in- tegrals of the form «.=/W, ". = ^W (22), ART. 5.] OF THE SECOND ORDER. 367 and if, regarding these as simultaneous, we determine^? and q^ as functions of a?, y, z, tliose values will be such as to render tlie equation dz =j)dx + fidy integrable, and thus to lead to the second or final integral. For simplicity, we shall represent u^—fiv^ by F^ and u^— (f> (^2) ^7 ^^' Thus the supposed first integrals are simply F=0, ^ = 0 (23). Now reverting to the system (18), and representing the ratio d?/ : dx by n, its first two equations assume the form, 1 _n-\-m _ nm R~~~S~ ~ ^ ' and shew that m and n are the two roots of the equation Hence, the value of the ratio dy : dx corresponding to one of the first integrals (23), is the same as the value of m cor- responding to the other. Now for the value of m corresponding to the integral F=0, we have by definition, duj^ dv^ dq __ dq du^ dp dp du^ dii^ dv, dv, dJ, dF -% dp (24). Again, seeking the value of the ratio dy : d.v, correspond- in o- to the integral <^I> = 0, we have 368 PARTIAL DIFFERENTIAL EQUATIONS [CH. XY. /d^ ^ (^^ ^ ]^ \dx dz ^ d]) del J d^ d^ d^ ^^ \ ^ _ 0 dj/ dz^ dp dq J " Equating the value of dt/ : dx hence found to tliat of m given in (24), we have, on reduction, dFd^ dF d^ dFd^ dFd^ dp dx dq dy dp dz ^ dq dz^ dFd^ /dF d^ dF d^\ dF d^ ^ dp dp \dp dq dq dp J dq dq In like manner equating the values of 7n corresponding to the integral <^ = 0, and of dy : dx corresponding to the in- tegral F=0, we have dFd^ dFd^ dFd^ dF d^ dx dj) dy dq dz dp^ dz dq dFd^^ /dFcl^ dFd^\dFd^^^^ dj) dp \dq dj) dp dq) dq dq Subtracting (25) from (26), there results dFd^_dFd^ dFd^_dFd^ dx dp dp dx dy dq dq dy flFd^_dFd^\ fdFd^ _dFd^\ ^ . ■^U-^ dp dp dzj^'^[dz dq dq dz J ^~ ^ '" ^■''>' Now this is identical with tlie equation (37), Chap. xiv. Art. 13, expressing the very condition which must be fulfilled in order that the values of p and q given by i^= 0, ^ = 0, may render the equation dz=pdx + qdy an exact differential. Hence the proposition is established. It is interesting to observe that the two first integrals stand in a certain conjugate relation. Each of them satisfies that partial differential equation of the first order and degree which AST. C] OF THE SECOND OliDER. 3G9 we should have to construct in attempting, by tlie process of Charpit, to integrate the other. Hence also, althouij,h the knowledge of Loth is desirable, that of eitlier is sufficient to enable us to proceed by integration to the final solution. 6. The statement of Monge's method, as derived from the above investigation, is contained in the following llule. Rule. The equation being Iir + Ss+Tt= V, form first, the equation Ilchf-Bdxdy^ Tdx^ = 0 (28), and resolve it, supposing the first member not a complete square, into two equations of the form dj — m^dx = 0, d?/ — in/Ix = 0 (-29). From the first of these, and from the equation Bd2Kli/+ Tdqdx- Vdxd?/ = 0 (30), combined if needful with the equation dz =pdx -i-qdf/, seek to obtain two integrals u^ — a, t\ = h. Proceeding in the same way with the second equation of (29), seek two other integrals w^ = «, v^ = /8, then the two first integrals of the proposed equation will be «:=/iW> ».=/.('■.) (31). To deduce the second integral, we must eitlier integrate one of these, or, determining from the two j) and q in terms of X, ?/, and 2, substitute those values in the equation dz — 'pdx + qdy^ wliich will then satisfy the condition of integrability. Its solution will give the second integral sought. If the values of w^ and m,^ are equal, only one first integral will be obtained, and the final solution must be sought by its integration. "When it is not possible so to combine the auxiliary equa- tions as to obtain two auxiliary integrals u = a, v = h, no first integral of the proposed equation exists, and some other pro- cess of solution must be soui2-ht. o B. D. E. 2i 370 PARTIAL DIFFERENTIAL EQUATIONS [CH. XT. We may observe tliat the determination of^:> and q from the two first integrals is facilitated by the fact that u and v satisfy the condition (7). Interpreted by Chap. ii. Art. 1, that con- dition implies that j? and ^ enter, in some single definite com- bination, into both u and v. Ex.1. Given ^,-a^^ = 0. Here R = l, S=0, T=-a\ F=0. Hence we have by (28) and (30), dif - a^dx^ = 0, dpdy — c^dqdx = 0 [a). The former of these is resolvable into the two equations dy + adx=Oj dy—adx=0 (5), of which the first gives y + ax = c, and at the same time reduces the second equation of (a) to the form dj) + adq = 0, of which the integral is ^ + a^' = ^' Thus a first integral of the given equation is p + aq = 6 {y + ax) (c). Proceeding in like manner with the second equation of {h), we find as another first integral 2:,-ari = '>ir{y-ax) {d). From these two equations determining p and ^, the equation dz =pdx + qdy becomes ^^^^(y+«^)+^(3/-Q^^) ^j. J /r(y-aa;) , ^ Ji 2i(Jj Or 7 _(j)(y + ax) [dy + ado^ — '^{y~ ^^) {^y ~ adx) Hence if ^ j"<^ (t) dt = , (<) and - ^ Jf (0 <^« =f. W, we have Here (jy^, -^^ are arbitrary functions since and '^ are such. ART. 6.] OF TPIE SECOND ORDER. 371 It is seen that, in each of the first integrals, the condition (7) is satisfied, and assuming 2:> + aq — (f) (^ + ax) = F, p — a^i — yjr {i/ — ax) = , it is easy to verify the condition (27). Ex. 2. Given r + as-\-ht= 0. Proceeding as before, we find J) + 7iq = (f){y — mx), p + mq = 'v/r (y — nx)^ as the two first integrals of the proposed, m and n being the roots of the equation f — at + h = 0. Hence, determining p and q, substituting in the equation ch —pdx + qdy^ integrating and reducing we have But when m and n are equal we have only one first in- tegral, viz. I) + mq = 0 (?/ — mx) . Treating this by Lagrange's process, we have the auxiliary system , dxi dz m 9 (3/ — i^ix) From the first two members we find y — mx — c. This enables us to reduce the equation of the first and third to the form , dz {ax +hy-\- cz) q = h(f) {ax + hy + cz) — a; and this must be integrated by Lagrange's process. The auxiliary system is, on representing ^{ax+hy+cz) by ^, dx _ _dy _ dz c c(f) h(j) — a' From these we find adx + hdy + cdz = 0, whence ax + hy + cz = C, and thus {ax + hy -\- cz) = (j) {C) . ART. 7.] OF THE SECOND ORDER. 373 Hence substituting dy= — {C) x = C', or 7/-\-x(f) (ax +h7/ + cz) = C. Thus the final integral is y + xcj) {ax + 1)7/ + cz) = ylr {ax -\-ly -\- cz). This solution may also be expressed in the form z = xcj)^ {ax + h7/ + cz) + yyfr^ {ax + !)?/ + cz) , in which it is in fact presented by Monge, {Application ds V Analyse d la Geometric^ Liouville's edition, p. 79). The equation solved is that of surfaces formed by the motion of a straight line which is always parallel to a given plane, and always passes through two given curves. 7. In the above examples V is equal to 0, and this always facilitates the application of Monge's method. The following is an example in which V is not equal to 0. Ex. 4. Given r—t = ^ x+y The auxiliary equations being dy"^ — dx^ = 0, djpdy — dqdx H ^ dxdy — 0, one of the systems hence derived is dy — dx=0, dp — dq-\ ^ dx = 0. There is also another system, but it is not integrable in the form u = a, v = h. From the first of the above equations we get y-x = a, ^I>-^^+^~r^ = ^y the latter of which may, since dz =pdx + qdx, be reduced to the form d{2y-a) {p-q)+2dz=0, 374 PARTIAL DIFFERENTIAL EQUATIONS. [CH. XV. whence {2y — a) {j> — q)+2z = h, or, replacing a'bj y-x, {x ^-y){p-q)+2z = h. Hence a first integral of tlie proposed equation will be {x-^y)['p-q)+2z=f{y-x), Now this being linear, we have, by LagTange's method, the auxiliary system dx _-'dy _ dz x-\-y~ x + y~ f{ij-x)-2z' The equation of the first two members gives y+x = a, and this reduces the equation of the second and third to the form — dy _ dz a ~ f{2y-a)-2z^ dz 2z _ f{2y — a) dy a a ' or •m r whence z = le "f{2y — ajdy-h h. The final integral will therefore be found by substituting in the above, after integration, y-\-xiox a, and F{y + x) for h, // 8. Monge's method fails in so many cases, owing to the non-existence of a first integral of the assumed form u =/(v), that it becomes important to inquire how its defects may be supplied. And various methods, all of limited generality, have been discovered. Thus Laplace has developed a method applicable to all equations of the form ^, >Sf, T, P, Q^ Z^ and JJ being functions of x and y only, — which consists in a series of transformations, each of which has the efiect of reducing the equation to the form s-\-F;p-VQ(i-\-Zz^U, ART. 9.] MISCELLANEOUS THEOREMS. 3.5 P, Q, Z and U "being functions of x and y, to which each transformation gives new forms. It may be that among these successive forms, some one will be found which will admit of resolution into two linear equations of the first order. But there are probably no instances in which this me- thod has been applied in which the solution may not be effected with far greater elegance, and witli far greater sim- plicity, by the symbolical methods of the following Chapters. And even Laplace's method is better exhibited in a symbolical form. The subject will be resumed. See Chap. xvii. Art. 14. The following sections contain miscellaneous but important additions. Miscellaneous Theorems. 9. Poisson has shewn how to deduce a particular iutegi'al -^ of any partial differential equation of the form P=[rt-syQ (45), where Pis a function of^, q, r, s, t, homogeneous with respect to the three last, n a positive index, and Q any function of X, y, z, and the differential coefficients of z of any order, which does not become infinite when rt — s'^ = 0. Assuming q_ = j>{p), we have 8 = 4,' {p)r, t = '(p)s={-(j>)Yr (46), values which make 7't — s^ = 0. Hence, substituting in (-io), the second member vanishes, while in the first, which is homoge- neous with respect to r, s, t, some power of r only will remain as a common factor. Dividing by that factor, we shall have an equation involving only p, ), and (j>'(p), i.e. p, q, and ~- . Integrating this as an ordinary differential equation we obtain a relation between p, q and an arbitrary constant; and this, integrated as a partial differential equation of the first order, gives the solution in question. Ex. G'lYen r''-f = rt-s\ 376 MISCELLANEOUS THEOEEMS. [CH. XV. Proceeding as aloove, we find 1 — {(/>' {p)Y = 0; therefore 1 - {{x,y, z,p, q)=0 (47) toe interchange x and p, y and ^, and change z into jpx -\-qy — z^ giving ^(p, q,px^-qy-z, x,7j)=0 (48); then, if either of these equations can he integrated in the form z = '\\r[x, y), the solution of the other icill he found hy elimi- nating X and Y between the equations _d^{X, Y) _ d^ (X, Y) '^~ dx ' y dY ' z = Xx-\- Yy-^lr{X, Y) (49). For, since dz —jpdx ■\- qdy, we have z —px + qy — j {xdp + ydcf) (50). Hence xdp + ydq is an exact differential. Represent it by dZ, and assume Z for dependent variable. Assume also two new independent variables X and Y, connected with the for- mer ones by the relations X=p, Y= q. Then dZ= xdp + ydq = xdX-\- ydY, ART. 10.] MISCELLANEOUS THEOREMS. 377 -rx dZ dZ Hence -j^=x, ~. = y, Z=j [xdp + ydq) =2JX + q7j-z by (50) ; therefore z =jpx + qy — Z= xX-\- yY— Z. On examining the ahove equations we see that x, y, z, an! X, F, ^are reciprocally related. Writing, side by side, the equations which are conjugate to each other, we have ■rr dz dZ '^~ dX' 3. dz dZ y-^dY'^ Z = Xx + Yy — z, z = xX-VyY- -Z. "We see too that the equations (49) which express one '^ct of the relations suffice to convert any relation found by inte- gration between X^ F, Z, where >^ stands for -v/r (X, F), into a corresponding relation between a?, y, z. Ex. Given z=i^q. Here the transformed equation is :px + qy-z= xy, of which the integral i^ z = xy -\- xf {-\ . Hence >/.(X, F)=XF+X/(J), and we have to eliminate X, F, between the equations x=Y Y,,fY\ JY\ ,. ., /I z = XF. 378 MISCELLANEOUS THEOREMS. [CH. XV. Each particular form assigned to / gives a distinct par- ticular integral. If we assume / f -^^ j = a -^ + &, we find from which, eliminating X and Y, we have z = {x — h){y — a), and this is one form of the complete primitive assigned in Chap. XIV. Art. 7. We may observe that the elimination may be so effected as to lead to general primitives. 11. 1)1 equations of the second order toe should have, m addition to the above transformations, to change f g y r into 2 > ^ ^'^^^ ~, 2 > ^ ^^^0 —^ 2 (51) J rt — s rt — s rt— s ^ ^ in order to form the reciprocal equation. Then the second integral of either being found in the form z = ylr{x, y), that of the other will be found as before by eliminating X and Y from (49). For since _dZ^ _dZ_ '^~ dX' y~dY'' therefore dx = RdX+SdY, dy = SdX ■\- TdY, . ,^ Tdx-Sdy j^ -Sdx-\-Bdy whence dX — -Dm_ ni -> ^.-^ = — 'FT— S^ — ' But X=p, Y=q, therefore , Tdx--Sdy dp = rdx + sdy = j^rp-S^ ' J ;j y ^j - Sdx-^Rdy dq^sdx^tdy= j^y_ ^2 > whence, equating coefficients, T -8 B ET-S'' '"'BT-S'' '"BT-S'' AET. 13.] MISCELLANEOUS THEOKEMS. 379 The extension of the theorem to higher orders involves no difficulty. 12. It is an immediate consequence of the above, that any equation of the form ^{lhq)r^it{p, q)s + xiP> q)t = 0 (52) can be reduced to an equation of the form X{^,y)r-'ylr{x,y)s + (j>{x,y)t = 0 (53), usually more convenient for solution. Legendre's solution of the equation by the aid of the above transformation, will be found in Lacroix (Tom. II. p. 623). The same transformation makes the solution of any equa- tion of the form Br+ Ss+Tt= V{rt - s') dependent on that of an equation of the form Br + Ss + Tt= F, but with different coefficients. The subject of these transfor- mations has been most fully treated by Prof De Morgan {Cambridge Fhilosophical Transactions^ Vol. Viii. p. 606). 13. Legendre also shews how, by a transformation for- mally resembling the above, to integrate the equation Assuming s and t as independent variables, and v = sx + f?/ — q as dependent variable, the equation is reduced to the form where S and T are the values of -r and -y^ furnished by the given equation. Lacroix^ Tom. ii. p. 631. 380 EXERCISES. [CH. XV. EXERCISES. 1. To what condition must u and v be subject, in order that u =f (v) may be a first integral of an equation of the form Br +Ss+Tt= V? Integrate by Monge's method the following equations : >■ 2. ccV + 2x^s + yH = 0. 3. q''r-2pqs+^H = 0. 4. Integrate jps — qr — 0. 5. Integrate by Monge's method the equation q{l+q)r-{p + q+2^q)s+p{l+p)t = 0. 6. The solution of Ex. 3 may, by the law of reciprocity, be made to depend on that of Ex. 2. 7. Monge's method would not enable us to solve the equa- 2p tion r — t = -^, X 8. Deduce by Poisson's method a particular integral of (1 + ^V - 2^5 + (1 +f) i = ^' 9. Shew that the equations rt-s'=:fip, q), and rt-s'={f{x, y)}'\ are connected by the law of reciprocity. 4:X 10. The solution of the equation r — t = (rt — s^) ^ p+q ^ ' mav be derived from that of the equation r — t-\ ^ = 0. •^ ■" x+y Art. 7, Ex. 4. ( 381 ) CHAPTER XVI. SYMBOLICAL METHODS. 1. The term symbolical is, bj a restriction of its wider meaning, applied more peculiarly to those methods in Ana- lysis in which operations, separated by a mental abstraction from the subjects upon which they are performed, are ex- pressed by symbols in whose laws the laws of the operations themselves are represented. Thus -J- is written symbolically in the form -7- u, the sym- bol -y- denoting an operation of which u is the subject. In thus expressing an operation by a symbol, in studying the laws of that symbol, and in founding processes and methods upon those laws, we introduce no strange or novel principle of Language ; for it is the very office of Language to express by symbols the procedure of Thought. Thus also we may write du ( d \ , , 5» + ''" = U+"J" ^^'' d\ du ^ ( d'^ d y\ ., ^^+«^ + ^'' = fe+"d.+T' ('^' and so on. It will be observed that the symbol precedes the subject on which it operates. Operations may be performed in succession. Thus d \/d U+^, ^. + ^1^^ denotes that we first perform on the subject u the operation 382 SYMBOLICAL METHODS. [CH. XVI. denoted by -j- + h, and then on tlie result effect the operation denoted by y^ + (^. Thus a and h being constant, we have — + ct] (-j~ -^h]u= ( -,- + a] (-^ + hu ax J \dx J \dx J \ax d fdu 7 \ fdu 7 dx \dx J \dx d^u , -..du 7 ,„. =&5 + («+^)^+«^'' (3). When an operation is repeated, the number of times which it is understood to be performed is expressed by an index attached to the symbol of operation. Thus d \(d =^+2«^+''" w- If in the second member of (3), as in the first, we separate the symbols from their subject, we have Now the symbolic expressions for the equivalent operations performed upon u in the two members of this equation are in formal analogy with the algebraic equation (m + a) [m -\-h) u= [m^ + {a + h) m + ah} u^ and tliis is a particular illustration of a general theorem to the statement and demonstration of which we shall now proceed. 2. If we compare the symbolical expressions xlx + «)(i + 0' £+('^+^)|+«^ («)' ART. 2.] SYMBOLICAL METHODS. 383 whose equivalence is stated in, (5), we see that each involves -J- together with constant quantities. Each might therefore, to borrow the language of analogy, be described as a function of -J- and constant quantities, or more briefly as a function of -^, and expressed in the form/fj-j. Again, each ex- presses a system of operations in the performance of which the presence of the symbol y- only indicates differentiation, not integration. We may with propriety term any function of -J- possessing this character a direct function of -y- . The theorem in question is then the following. Theorem. Any direct function of -j- and constant quan- tities may be transformed as if -j- were itself a quantity. In the first place it is evident that any direct function of the symbol -7- according to the above definition is, in form, what we should term a rational and integral function of -j- , were that symbol merely algebraic. Now the laws, according to which algebraic symbols com- bine witli each other in the composition of all rational and integral expressions, are the following, viz. Ist, the distributive, expressed by the equation m {u-\-v) —mu+ mv (7), 2ndly, the commutative, expressed by the equation ma = am (8) , Srdly, the index law, expressed by the equation 7wW = m"-^^ (9). 384 SYMBOLICAL METHODS. [CH. XVI. These determine, and alone determine, the forms, or, to speak more precisely, the permitted variety of form, of algebraic expressions of the above class. But the symbol j- , when employed in combination with constant quantities to operate on subjects which are not con- stant, is subject to laws formally agreeing with the above. For we have }ju + v) = ~u+^^v (10), t dV/dY (d b / J\a+i U i^T^ a^), dx) \dxj \dxj the last of these, however, expressing, not any distinctive pro- perty of the operation -v- , but only the fact that it is an operation capable of repetition. These laws, in like manner, determine the possible forms of symbolic expressions involv- ing -J- with constants, and representing direct systems of operations. Hence the variety of form permitted in the one case is the same as that permitted in the other. In other words, the same transformations are valid. Among the consequences of the above theorem the following may be noted. 1st, We can reduce any symbolical expression of the form j;^.- + «i^;^ + a2^^^, ...+«„, in which a^, a^,...a„ are con- stants, to an equivalent expression of the form d \ f d \ f d ='' ('•^)- Then on the above principle of notation we should have or, with not less propriety of expression, 1 u = dx'' ' 'dx^' tiie last two equations differing in Interpretation from (16), not at all as toucliing the relation between u and t% but only as more distinctly presenting u as the object of search. Of what avail then, it may be asked, Is that analogy upon which the expression of the last two equations is founded? If a convention, it is at least a very natural one, that we should express an operation performed upon a subject, by attaching, in some way, the symbol denoting the operation to the symbol denoting the subject. The order of writing, in that family of languages to which our own belongs, has 388 SOLUTION OF LINEAK EQUATIONS [CH. XVI. doubtless determined tlie mode of connexion actually adopted, and wliicli is the same as if the symbol of operation were a symbol of quantity employed as a coefficient or multiplier. It comes to pass, moreover, that the formal laws of combina- tion in the direct cases investigated in Art. 2 prove to be the same for the symbol -j- as for a coefficient or multiplier. But inverse symbols derive their meaning from the direct operations to which they stand related: they are forms of interrogation, the answers to which are to be tested by the performance of the direct operations. Hence it may be inferred that the laws for the transformation of inverse, expressions involving — (XX with constants will be the same as for the corresponding forms of ordinary algebra. The analogy consists, not in the mere adoption of a common notation, but, as all true analogy does, in a similitude of relations. If the equation [-^ a ) w = X be given, we have 4. Solution of Linear Equations loith co7istant Coefficients, A dx but, the known general solution of the given equation being U = 6 we see that ^^ «) 'x^e-^je-^'^Xdx (17), \dx an arbitrary constant being introduced by the integration in the second member. If X= 0, we have i-«r-^=c/.- (18). These results we shall have occasion to refer to. ART. 4.] WITH CONSTANT COEFFICIENTS. 389 Ex. Xow suppose the given equation to be d^u , ^. du we have, on separating the symbols, or, by Art. 2, (i-«)(i-^)»=^^' (^')' Hence [^ _ 6j « = (^£ -.) X, (i-f(i-r^ « On comparing this with (19) we see that, in inverting a system compos|ed of two operations performed in succession, the order of the ' operations themselves is inverted. This is evidently true whatever may be the number of successive operations, the last to be performed being always the first to be inverted. Fi'om (20) we might deduce the actual value of u by suc- cessive applications of (17). Such was the method once em- ployed. But it is better to proceed as follows. From (19) we have E-)(a-C-^' '^■'- No^v by the known theory of the decomposition of rational frac tions {(m - a) (m - b)}-' = N^ {m - a)-' + ^, {m - IT'...^ (22), iV"^, ^^2 being functions of a and h, which may be determined in various ways, but most directi?/ by multiplying both sides of the', equation by [m — a) {in — h), and equating coefficients. 390 SOLUTION OF LINEAK EQUATIONS [CH. XVI. Now the suggested transformation of the expression for u given in (21) is d \fd dx J \dx -^)r^=^'(£-^)"'^+^'"<£-'r'''-(^^)- And, from the very definition of inverse forms, the proper test of the validity of tiiis transformation is, that the performance of the direct operation I- ^) ( t — ^) on the second member shall reduce it to X, Effecting this operation, and remembering in so doing that -j a and -^ h are commutative, and that by definition a) (-^ a\ X= A', the second member becomes or (iv;+i^gg-(5.^^ + aiv;)x (24), and this reduces to X if .V, + -V, = 0, hN, + aN=-l (2iS) \ But these equations for the determination of N^ and A^g ^^® the same, and necessarily the same, as we should have fecund by multiplying, as above indicated, (22), by {in -a) (m—b), and equating coefficients. The two series of operations only differ in that -p occupies in the one the place which m c Occu- pies in the other. Determining Nj_, N^, we see that u may be expressed in the form d V'./d •-' • ^ a — O] \dx «r^-(£-^r^} ^•^«)- ART. 5.] WITH CONSTANT COEFFICIENTS. 391 Hence, by (17), and as, on effecting the integrations, two arbitrary constants will be introduced, this is the most general value of u. 5. In like manner if there be given the general linear differential equation with constant coefficients (£+^'£^+^=£^-+^«)''=^ c^^)' and if we represent by a^, ^g, ... a„ the roots, supposed all dif- ferent, of the algebraic equation mr + A^nr'-\-A^m''-'\.,+A, = 0 (29), then the given equation may be expressed in the form (£-"') (i-"J-(;^-"")" = ''^^' whence "-{(e-°0(e~"-)''-(;s-"-)}'^" -;''.(i-".)"'-^-.G4-.r---(i--n-^--^-«- the decomposition in the second member formally resembling that of the rational fraction. If the equation (29) have r roots equal to a, there will exist in the resolved expression for u a series of terms of the form h(i-r-Mi-"r'---(£-r)-^-^^-<">' • This theorem was first published in the Cambridge Mathematical Journal (1st series, Vol. ii. p. 114), in a memoir written by the late D. F. Gregory, then Editor of the Journal, from notes furnished by the author of this work, whose name the memoir bears. The illustrations were supplied by Mr Gregory. In mentioning these circumstances the author recalls to memory a brief but valued friendship. 392 SOLUTION OF LINEAR EQUATIONS [CH. XVI. or, which is preferable, a single term of the form A^Bi^ci,...^n^Ml-A"x (32), dx dx^ ' ' ' dx^ V \dx A, B,...R being determinate constants. Now since, by (17), i^- S^ X=e''''{ e-'''' Xdx-, therefore (^ " «) ' ^=' (^ " ^) ' ^"/^"" ^^^ = e- L-- {e^^ L-- Xdx) dx ^ e"' [L--- Xdx\ Proceeding thus, we have Ex.l. Given ^'! + 4'^ + 3^-4^^-4y = X dx dx dx dx '^ This equation gives, on decomposing the complex operation performed on ?/, Wore . = {(i-^^y(£.l)g-l)rx. Now, ^1 ^^ ' =±!^ ^+ 1 w-f2)'(w+l)(m-l) 9(w+2J' 2(m+l) l«(m-lj ' Therefore 9V rfj; 4 ;- + 1 1 1 e^'ili'Xdx'- - i 6-'je'XcIx + -€' j e"Xdx. ART. 6.] WITH CONSTANT COEFFICIENTS. S93 Ex.2. Given P^,+7l'u = X. ax Here w = ( ^r-g + w^ ) X. But e"^V(-i)fe-"V(-^)XJaj = [cos 7^a; + \/(— 1) sin wa;} ] I cos nxXdx — V(— 1) 1 sin nxXdx\ = (cos ??a; - V(- 1) sin nx\ \ \ cos nxXdx■\^^J{-^) \ sin ?iajA7Za:^ , whence, on substitution and reduction, u = - J sin nx \ cos nxXdx — cos tioj I sin nxXdxY . 6. When the second member X is a rational and integral function of x, the final integration may be avoided. For, representing the given equation in the form fi-j-] ic — X-\-0^ we have HAW^MBY" <"'• A particuhar value of the first term will be obtained bj de- veloping ]/(^)r in ascending powers (so to speak) of -y-, and then performing the diftercntiations on A" while the general value of the second term will introduce the requisite number of arbitrary constants. 394 SOLUTION OF LINEAR EQUATIONS [CH. XYL 72 Ex. Given -j^ + 71%= l+x + x^. Here « = g+„f (1 + . + .=) + g+„^ro + C^ cos 71CC + C^ Bin «.r = 7i~^ {1 + X +x^) — 2?i~* + Cj cos nx + Cg sin 7ia;. The validity of the transformation of the inverse form {'T~i-^^^] ^7 development, as of its other transformation by decomposition, is tested by performing on the result the direct operation ~p-r^ + n^. We take occasion to notice that different transformations, while equally valid, do not of necessity con- duct us to solutions equally general, nor have we any right to expect that they should. Each solution is an answer to the question contained in the given inverse form, but that question may admit of different answers, and no solution is general which does not include them all. The final integrations may also be avoided when X consists of a series of exponentials of the form e*""" with coefficients which are either constants, or rational and integral functions of a?. Since (-j-\ e"'"' = ??2"e"*'', we have, for all interpretable forms ofyf-7- j , the relation the second member expressing the complete, because the only, value of the first member when/ (-7-) is rational and integral, AET. 6.] WITH CONSTANT COEFFICIENTS. 39o but a particular value of the first member when /'(-^ J is inverse, the test being as before. Hence, if the given equation be/( t-) u = SA,,,€^'', we have dxj dxj\ ^ " ' K \cl IX = S^„.(/Wl-V'"'+j/g)fo (36). the second term introducing the requisite number of arbitrary constants. Again, if, in any expression of the form /i 7- 1 e'^^A", we convert -j- into — + -j^ , where -j^ operates on x only as ax ax ax ax ^ contained in e"*^, and -7^ operates on x only as contained in A", we have Hence, dropping the suffix which is no longer necessary, since A' alone follows the operative symbol, we have /(S^"'^^'=^-V(;J + -)^l' (37). When therefore X is a rational and integral function of x, a particular value of the first member may be found from the 396 SOLUTION OF LINEAR EQUATIONS [CH. XYT. second, by developing the functional symbol and effecting the differentiations. And that particular value may be made general, as in the following example. Ex. Given -^ - 3 -y- + 2i* = a?e'"^ ax ax I (m - 1) (to - 2) 1(to - 1) («j - 2) J' dx ^ J cce-' (2TO-3)e''" (to - 1) (to - 2) [(to - 1) (to - 2) j'' "*" ' "^ 2 ■ Again, the theorem (37) relieves us from any diiSculty arising from cases of failure referred to in Chap. ix. Art. 9. Es. Given (^-aXu = e". c.r X € 4 c^-\-c^x... + c^^x + — '2...n AVhen the second member X involves terms of the form Ao^o^mx, Bsinmx, &c., we may either substitute for them their exponential values, or we may employ directly the easily demonstrated theorem f d^\ sm ., 2. sm -r— 7nx = / (— m) mx, \ax J cos '^ ^ ^ cos ART. 6.] WITH CONSTANT COEFFICIENTS. S97 Ex. Given y^ + w^w = Sa„i cos [mx + ^) . Here u = f y^ + n^\ Xa,^ cos (ma; + ^) + f yr, + 7iM 0 ^ «„, cos (mx +h) , ^ , n • = 2,-^ ^-^ r, -+ C, cos 72X + C\ Sin Tiaj. ?r — m In this example, however, the failing case which presents itself when m = n, is most simply, though not most satis- factorily, treated by the methods of Chap. ix. Art. 11. The reduction of an integral of the 7i^^ order by the fore- going theory is not devoid of elegance. We have //...X.." = \dx. Now letiz ; = e^, then dX dx- .,dX -' de' d^X dx' -,d . -'^'de' .,dX dd = e' Proceedin g thus, ■ we have dx"" Z = e-"^ [d -) [dd |-l)|A',l>y(37). .,4X..(38), and therefore the operation denoted by [^-) , and the com- pound operation denoted by -^''--oa— ^)-i [de are absolutely equivalent. Hence inverting both, and observ- ing that the inversion of tlie latter involves the iuvcrsiou ot the order of its component symbols, we liave 398 FORMS PURELY SYMBOLICAL. [CH. XVI. dx )--&c.} ^(»^^)_^g_^^3,_^^_^^.^ 1 (,,n-.,.,^<^N"' = ^- • L"-' lxdx-in-\)x"-^ \Xxdx 1.2. ..(«-!} i J ^ ' ) the result In question. From (38) we have the theorem which is important in the transformation of differential equa- tions. Forms purely symbolical. 7. In any system in which thought is expressed by sym- bols, the laws of combination of the symbols are determined from the study of the corresponding operations in thought. But it may be that the latter are subject to conditions of possihility as well as to laws ivhen possible. And thus it may be that two systems of symbols, differing in interpretation, may agree as to their formal laws whenever they both express operations possible in thought, while at the same time there may exist combinations which really represent thought in the one but do not in the other. For instance, there exist forms of the functional symbol /] for which we can attach a meaning to the expression f{ni), but cannot directly attach a meaning ART. 7.] FORMS PURELY SYMBOLICAL. 399 to the symbol /(^) • And the question arises : Does this difference restrict our freedom in the use of that principle which permits us to treat expressions of the form /(-r-j as if ,- were a symbol of quantity ? For instance, we can attach no direct meaning to the expression e '^f{x), but if we de- velope the exponential as if y- were quantitative, we have =zf[x + h) by Taylor's theorem. Are we then permitted, on the above principle, to make use of symbolic language ; always supposing that we can, by the continued application of the same principle, obtain a Jitial result of interpretable form ? !Now all special instances point to the conclusion that this is permissible, and seem to Indicate, as a general principle, that the mere processes of symbolical reasoning are independent of the conditions of their interpretation. In the few Instances we may have occasion to employ, verification will be easy. We take occasion to notice that, whatever view may be taken of this principle, whether it be contemplated as belonging to the realm of a priori truth, or whether It be regarded as a generalization from experience, it would be an error to regard it as in any peculiar sense a mathematical principle. It claims a place among the general relations of Thought and Language. On the principle above stated we should have ^'^'''V(-r,2/) = / = /V(-r,2/) 400 FOEMS PURELY SYMBOLICAL. [CH. XVL And here, the expression e '^'^ ^^ , which is without meaning in itself, is to be regarded simply as the representative of the expression which has meaning. And the proper test of the validity of the symbolic equation dx dy __ dx dy consists in substituting for each exponential form the series it represents, and comparnig the finally developed results, just as we shovild, by developing the exponentials, verify the alge- braic equation, It must be noted that -^ and -y- are commutative, and ax ay combine, in all respects, like symbols of quantity. We are not a:+-- — d permitted to write e ''■'= e^'e'^'', because x and -y- are not commutative. 8. The above principle is illustrated in the solution of the following partial diiFerential equations. . d\i ^d\ . . Ex. Given ;^. - « j^2 = ^ (^: V) • IIerez.= (|^-a^|.) [x,y) ^ Ua -^y |e"^^|e~"^^^(^ {x, y) dx-e''''^^jr^"<}>[x,y)d^ AKT. 8.] FOKMS PURELY SYMBOLICAL. 401 the forms of j and ^^ being given by tlie equations *^i i^y y) =\ ^ {pc,y-ax) dx-Y'y^r {7/), ^2 (^^ y)=ji'{^,y + O.X) dx + x (y)> -y^ {y) and ^ iv) being arbitrary functions of y. If (f) {x, y) = 0, we hence find or, if we represent ^j^(^)% ^7 ^1(3^). and ^fx(y)(^!/ ^y Xi ii/)> ^ = fAu + «^) + xAy- «^)- As -v/r and X ^^ arbitrary, -v/r^ and Xi ^i'^ so too. This agrees with the result on p. 370. T^ ^. d'^u d^u d~u ^ Ex. Gmn^ + ^^.+;^ = 0. We may put this in the form y-^ + ait = 0, where a stands d^ d^ . for -^ + -,^2 5 a^^^ integrate wdth respect to x, as if a were a constant quantity. Eemembering that the two arbitrary con- stants of the complete integral must then be replaced by two . arbitrary functions of y, z^ we get the symbolical solution Developing the cosine and the sine, and replacing (d' d'\h , . . by a new arbitrary function x {l/j ^)j ^^'^ have B.D.E. 26 402 GENERALIZATION OF THE [CH. XVI. x' fd' , d'V , , . Under this form, the solution is presented by Lagrange in the Mecainque Analyiique, Tom. ii. p. 320. Generalization of the foregoing theory. 9. All equations, whatsoever their nature or subject, which are expressible in the form [ir''+Ay-'-\-Ay-\.. + A,:)u=^X (1), where it is an operative symbol subject to the laws irau = airu, ir {u + v) = 7ru + ttv, ir^ir^'u = tt"'^'' w, a being a constant and u and v functions of x, admit of trans- formations analogous to those of Art. 5. Thus, since w = (tt" + Ay-^ + Ay^ . . . + A^^ X, we shall have, when the roots a^, a^^...a^ of the auxiliary equation Tif + A^m""-^ + A^m""-'' . . . + ^„ = 0 are real and unequal, the transformation u = N,{iT-ayX-^N^{ir-ayX..,-rK[^-aX'X...{^^\ the coefficients iV^, N^^ ... N^ being determined as in Art. 5. The legitimacy of this transformation is proved by operating on both sides of (2) with Tr^' + ^jTr""^ ... + ^n, and shewing ART. 9.] FOREGOING THEORY. 403 that (1) is reproduced with the same conditions for deter- mining iV^ , iVg , . . . iV^„ as if TT were a symbol of quantity. But the question of its completeness, of its conducting, through the performance of the inverse operations (tt— aj~\&c., to the mo^i general solution of (1), is one that we are not called upon to determine a priori. In all the cases we shall have to con- sider, its completeness will be obvious. Ex. The equation -j^,- {2x^-1) ~-\- {a? + x-l) u^O is reducible to the form tt {ir — \) u = 0 where 17 = ~j — x. Hence u= (7r-l)''0-7r-'0. Let (tt — 1)~^ 0 = y, then, since (tt — 1) ?/ = 0, we have dx ^ + l)y = -0, (x+l In like manner, if if ^0 = zr \YQ find dz dx ■xz = 0, Z a:2 Therefore a;2 A very interesting application of the same theory to the solution of partial differential equations is afforded by what Mr Carmichael has termed the index symbol of homogeneous functions. Cambridge and Duhlin Math, Journal^ Vol. yi. p. 277. Since, if u^ represent a homogeneous function of the a^^ degree of the variables ic^ , o^g , . . . a:„ , we have dua du„ du. ,„. "'■^:+^^&r ••■+-» i^:=-« (3)> it follows that, if we represent the symbol x^ -vy ... + ^\'j~^ by IT, we shall have iru^ — au„ , 7ru„ = a^u^ , &c. 2G— 2 > 404 GENERALIZATION OF THE [CH. XYI. and therefore, in accordance witli the reasoning of Arts. 3 and 4, /(7r)Wa=/Ww« (A an equation of wliicli the second member expresses the com- plete, because the only, value of the first member when/(7r) is rational and integral, but a particular value when the first member contains inverse factors. Hence, if we have any equation /(tt) u = X^ where /(tt) is of the form tt^ + A^'^ + ^y' ... + A^, and X is a series of homogeneous functions of the variables, suppose X=X„+X,+ ...&c., we get « = {/W}-^Y+{/(,r)l-0 = {/(,r)}-> a; + i/w)-' X, ... + !/w}- 0 = {/(«)}-' X„ + {/(i))-' X, ... + i/Wr 0, by [A). To find the value of the last term, we proceed, as in Art. 5, to reduce it to a series of terms of the form ^^(Tr — a)~*0, * being the number of roots equal to a of the equation /(7/i) = 0. Now it may, by an induction founded on successive applica- tions of Lagrange's method for the solution of linear partial differential equations of the first order, be shewn that (TT-a)- 0 = u, (log x^-' + v, (log x^-\.. + ic,.„. (B), Uai v„, Wa being arbitrary homogeneous functions of x^, x^, x^ of the a^^ degree. To this result we may give the symmetrical form (tt - a)-' 0 = u,L'-' + vJP~' . . . + 2^„ , L, M, &c. being logarithms of any homogeneous functions which are not of the degree 0. It remains to shew how it may be ascertained whether a propo.sed partial differential equation can be reduced to the ioim f {tt) u = X, ART. 9.] FOREGOINa THEORY. 405 Let us resolve each symbol y- , entering into tt, into two, and let -,— represent -7- as operating on x^ only as entering into M, and -j- only as entering into tt. Also let It is easily seen then that tt = tt' + tt". We have therefore ITU = (tt — tt") u = iru ; therefore '7r''^u= (tt — tt") ttu (C), But as tt", in (0), operates on the variables only as entering into TT, which is a homogeneous function of those variables of the first degree, we may replace it by unity. We have there- fore tt'^u = (tt — 1) TTU. In the same way it may be shewn that ir'^'u = (tt — r + 1) (tt — r -f 2) ... ttu. And thus it is seen that any partial differential equation which is expressible in the form /(tt) w = X, on the hypothesis that -p- , j— , &c. operate on the variables only as entering into u, is reducible to the form (f) {it) u — X, independently of such restriction. This reduction having been effected, the solution can be found by means of [A) and [B]^ whenever the second member con- sists of one or more homo^reneous functions oi x^.x,. ..,x . O 1 ' 2 ' H Ex. ^ — + 2xy^^^+y ^^,-n{x^ + ,j^J^„u = x- + i/ + x^ Here we have (tt'^ — mr' + n) u = x^ i-y^ + x^. Therefore {tt (tt — 1) — wtt + n]u = x^ + t/^ -\- x^j or (tt ~ w) (tt — 1) w = aj^ +2/" + ^^i 406 GENERALIZATION OF THE [CH. XVI. whence t, = {(tt - n) (tt - 1)}-^ [x^ + f + x'] + {(tt - n) (tt - 1)}"^ 0 «^^ + / . -^^ ...^, (2-n)(2-l) (3-?i) (3-1) Un , v^ denoting arbitrary homogeneous functions of the degree n and 1 respectively. 10. We may, by simple transformations, reduce to the above case various other classes of equations differing from the above only as to the form of tt; e.g. the class in which rj^^a^x^-^-Va^x^-^ ... + a„a;„^; but, passing over such special forms, we shall consider the general equation/(7r) u = Z, where and each of the coefficients X^, X^,...X^, as well as Z", may be any function whatever of the independent variables. And we design to shew, first, how it may be determined whether a given equation admits of reduction to the more general form above proposed; secondly, how, then, to integrate it. Suppose the given equation of the n*** order; then the symbolical form in question, should the proposed reduction be possible, will be Now the highest differential coefficients in the given equation will arise solely from the symbol tt", and the terms in which they occur will enable us to determine the form of tt. Thus, for two variables, we have \ ax ay J ax dxay ay (ypdM ^dM\du (jifd^ ^dN\du \ dx dy J dx \ dx dy J dy^ ART. 10.] FOREGOING THEORY. 407 in which the terms containing: -7-^ , -^ — 7- , -— , are the same ^ ax dxdy dy as they would be, if, in the first member, -j- , -7- were sym- bols of quantity. And this law is general for the highest differential coefficients. Again, the form of tt being determined, the values of A^, A^, ... will, whenever the proposed reduction is possible, be found by effecting the operations implied in the first member of (4), and comparing with the first member of the equation given. Suppose the equation reduced to the form (4). Then, if the auxiliary equation m"" -h A^mT-' + A^m""^ ... + A^ = 0 (5) have its roots all unequal, we have a series of terms of the form (tt — a)~^X; and each such term involves the solution of a partial differential equation of the first order of the form ^^ du ,^ die ^ du ^ But, if the auxiliary equation (5) have equal roots, partial differential equations of higher orders will present themselves. We deem it therefore important to shew how this difficulty may be avoided, or, to speak more precisely, how its solution may be made to flow from that of the corresponding case of linear differential equations with constant coefficients. Introduce a new system of independent variables ^i , 3/3 , . • -^n » so conditioned as to give tt = -v- . To prove that such a sys- tem exists, and to discover it, let us assume y^, y,^, ... y„, in succession, as subjects of the above symbolical equation, and examine whether the results are consistent. And first, assum- ing yi as subject, we have 408 GENERALIZATION OF THE [CH. XVI. Secondly, assuming ?/i, representative of any of the remain- ing variables y^, y^, -" Vn^ ^^ subject, we have the equation ^•i;-^^t -^^"t=o (^)- It follows from the above that, if we integrate the auxiliary system ' X-X"'~~^n • ^^' the values of y^, y^,...y^ will be the first members of the integrals of that system expressed in the form And it follows from (6) that if, from the system dx dx^ ^ , . . X, x/- X, "^^^ • ^^^^' differing from (8) only in that it contains one additional mem- ber dy^^ we deduce an additional integral equation connecting y^ with the original variables a?^ , x^^ ... ic„, that equation will give the value of y^. We see that the number of distinct auxiliary equations is precisely equal to the number of quan- tities to be determined, so that the scheme is a consistent one. The solution of the problem is therefore virtually dependent on the partial differential equation (6), from the auxiliary system of which, (10), it suffices to deduce n integrals, one expressing y^ in terms oi x^^ x^^ . . . x^, the others determining y^^ yz^ •'• yw) ^^ functions ot'.Tj, x„... x^, in the forms (9). To the arbitrary constant in the value of ?/^ we may give any value we please. Introducing the new variables, the equation given now as- sumes the form /(^)^*=(yi.y2v2/«)» which must be integrated as if u and y^ were the only varia- bles, an arbitrary function of y^, y^, ... y^ being introduced in the place of an arbitrary constant. Finally, we must restore to 2/i , 2/2 > • • 'Vn tl^eii' values in terms of a?^ , x^, ... a?„. ART. 10.] FOREGOING THEORY. 409 Ex. Given (1 - x'f ^' + 2 (1 - x') (I - xy) ^"^ + il-xyY^^-^x[l-x^) J - {x+y-ix^y) | +,™=0. Here, the form of the first three terms shews that we must have TT = (1 — x'^) -7- + (1 — ^y) -j- , and the equation assumes the form (7r'+?i')zf = 0. To avoid the difficulty arising from the imaginary factors of TT^ + n^j let us assume two new variables, x and y', such that we may have tt = -7-7 • Then by (10) corresponding to which we have the integral systems y-x , , //l + a;\ , Hence, if we assume , , /fl-\-x\ , y-x we get the transformed equation therefore u = cos nx'cj) [y) + sin 7ixyfr(7j)j or, restoring to x and ?/' their values, — ~'{"'»S'\/(l'^)}*V(Sj 410 EXERCISES. [CH. XYI. EXERCISES. 1. 5^_2^ + ^ = 6^ dx^ dx^ dx dx dx 3. Determine the solution of the above equation when dx dx d^u ^du , ^ 5. -7-^ 4- 3 -7- + 2w = cos Twa;. OtiC ctx / d y* 6. Solve the equation (-^ a) u = cos mx. In the above example it will be most convenient to proceed thus : COS mx + « (if" (S-) = ^ — ^ f^+a] cos mx+e^ici + c^... -\-c„2ir-^). 7, Solve the equation f-j aj u — e' cos mx, ^dj'u ^ d'u , ^d'u / du ^ du \ ^d^u d^u . 2^'w , a , oJ CH. XVI.] EXERCISES. 411 ^y^ 10. Solve, hy the method of Art. 10, the equation / d d dV 2 ^ 11. The solution of any equation of the form d^u ,^ ,^ . du fdX ^„ „ , \ may be reduced to that of two linear equations of the first order. ( 412 ) [CH. XVII. CHAPTER XYII. SYMBOLICAL METHODS, CONTINUED. 1. The classes of equations considered in tlie last chapter might all be gathered up into the one larger class represented f{ir)u = X, 77 being a symbol combining with constant quantities as if it Avere itself a symbol of quantity. But linear differential equa- tions do not, except under particular conditions, admit of expression in this form. Those which are of the ordinary species involve in their general expression two symbols, x and y-, operating in combination on the sought and dependent variable y ; and no substituted form of such equations is general which introduces fewer than two symbols in the place of X and y- . We propose in this chapter to employ a trans- formation which is general, and which is adapted in a very remarkable degree to the development of general methods of solution. A somewhat fuller account of it will be found in a memoir on a General Method in Analysis {Philosophical Transactions for 1844, Part II.). Other principles and other methods will also be noticed. The following theorems, demonstrated in Chap. xvi. will frequently recur. f I If ic = e^, and if -7-, be represented by Z), then x'^^ = D{D-\)...{D-n + l)u (1), while the relations connecting ^ and e^, become (( ART. 1.] SYMBOLICAL METHODS, CONTINUED. 413 f{D)e-'=f{m)e-' (2), f{I))e-'u=e-'f{n + m)u (3). The latter of these relations enables us to transfer the ex- ponential e"*^ from one side of the expression /(2>) to the other, by changing D into I) ± m, according as the transference is from right to left or from left to right. Thus, as another form of (3), we should have e-'f{D)u=f{D-m)e-'u (4). It is an immediate consequence of the above theorem that eveiy linear differential equation ichich can he exjyressed in the form, (« + 5a; + c^^..)J+(a' + J'a; + cV...) j!!^+&c. = X...(5), can he reduced to the symholical form, UD)u+fSD).'>a+f,{D)e'u + &c. = T (G), where T is a function of Q, For, multiplying the given equation by cc", and assuming ir = e^, the first term of the left-hand member becomes, by (1), and this is reducible, by (4), to the form aI){I)-\)..,{I)-n\\)u-\-h{J)-\){B-2). .,[!)- n) e'u + c[D-2)[D-Z)...[D-n-\) e'u + &c., each term of which is of the general form - 2) e'^u, whence, substituting, and collecting together terms like with respect to the exponentials, we have D{D-l)u + [a [D - 2y ± n'] i' u = i'j> {/) as the symbolical form. To return from the symbolical to the ordinary form of a differential equation, we must, by (3), transfer the exponentials to the left of each symbolic function f{D), convert the latter into a series of factorials of the form D [D - 1) ... (D — ?2 + 1), and then apply the transformation (l). Ex. 3. Given D [D -I) u-\- D [D + 1) eH^ 0. AYe have in succession, Therefore, dividing by x, A symbolical equation which has only two terms in its first member may be termed a binomial equation ; one which has three terms a trinomial equation, and so on. We may deter- mine by inspection to which of these classes an ordinary diflferential equation is reducible. For multiplying it by such a power of x as to permit its expression in the form ART. 2.] SYMBOLICAL METHODS, CONTINUED. 415 where A, B, &c. are algebraic polynomials with respect to x; the number of distinct powers of x involved in those polyno- mials will determine the number of terms in the reduced symbolical equation. Ex. 4. Thus the equation (a ^hx)-^, + {c+ ex) ^ + qu = 0, being expressed in the form (a + hx) x^ ^—2 + {(^^ + ^^'^) ^ ;7~ + iq^^) ^j it is seen that its symbolical form will be trinomial, since the terms within the brackets involve x in the degrees 0, 1, and 2. Finite solution of differential equations expressed in the symbolical form. 2. If we affect both sides of the symbolical equation (6) with {f,{B)]-\ then for f,[D)y^{D) write (/>,(!)) &c., and for {/„(i>j}"^ r write U, we shall have u + ci>^{D)e'a-\-cf>,^{D)e-'u...+ ci>,^[D)e^hi=:U (7); and under this form the equation will be discussed in the fol- lowing section. Pkop. I. The equation u + «,(/) (Z>) ehi + a^(/) (D) (/) (i) - 1) e^^^^ . . + a„(/)(i))^(i)-l)... [D) e^a = U, the values of q heing determined hy the equation 2" + «,f"^ + «,f-^... + a„ = 0. 416 FINITE SOLUTION OF DIFFERENTIAL [CH. XVII» For ^ {D) - 1) e'hi = cj, [D) e'4> (D) e'u = { {!)) e']\ and in general ^{I))cp{D-l)...cl>{I)-n + l)e^'^ii= {(/) {D) 6^}"^i. So that, if we represent tlie symbol cj) [D) e^ by p, the equation in question becomes therefore u = [l^-a^p + a.p' . . . + a„p")"' TJ = {iv; (1 - + 1) (i> + 2)w-3(Z) + l) [D-2)eht + 2 (Z> - 2) (Z) - 3) e-hi = ae"^ whence D + 2 ' {D + 2){U + 1) („ + 9)(,i + i)' or, puttms: -r, ^ = Pj "t ;:^7 Tn = -^i ' ^ "= i> + 2 ^' {n + 2){n + l) ' (1 - 3p + 2p'o ?^ = r. Hence w = - — t-^T={ - — — - "1^ 1 - 3p + 2p~ \l-2p 1-pJ = 2u^-u^^ («)» where v^={l-2pyT, u, = {l-prT, B. D. E. 27 I 418 ^^ -^ SOLUTION OF DIFFERENTIAL [CH. XVII. From the former we have I) — ^ ae^^ whence {D + 2)u^-2 {D -2) e\= -^^; and this gives (,_2.^)5 + (2 + 2.),, = ^ (^)- In like manner we find, for Wg, ,(.-.=)J + (2+.)„, = ^ W- The values of u^ and z/g* determined from (h) and (c), and substituted in {a), will give the complete solution. If a = 0, we find u _ c^{i-2xy+c,{i-xY X 3. We proceed to consider more fully the theory of the binomial equation u + ^{D)6'^u=U (10). Peop. II. The equation u+ cj) [D) e^hi = U ivi'll he converted into V + (j>{D + n) e^^v = F, hy the relations For assume u = e^^y, and, substituting in the original equa- tion, we have therefore e^'^v + e"^^ [D + ?i) e^^y = C/; by (3), Let Z7=e"^F; then the above becomes as was to be shewn. ART. 3.] EQUATIONS IN THE SYMBOLICAL FORM. 419 Thus in any hmomml equation ice can convert (f> (D) into (j){D + n), n heing any constant. Prop. III. The equation u + (^{D) e^u = U vnll le converted into V + -^ (Z)) e^^v = F", hy the relations, where P^ . denotes the symbolical product ^{D) (f> {D - r) (j) {D ^2r) ... f{I))fll)-r)ylr{D-2r) ...* For, assume u =f{B) v, and, substituting in the original equation, we have f{D)v + 4>{D)e^J{D)v=U; therefore f{D)v + 4> {D)f{D - r) e'«y = U, by (4) , ,+i(M^...„={/(z))rr .(11). Comparing this with the equation v +^ {D) e^^v = F, we have therefore /(i)) = ig)/(i)-,.). Hence fiD-r) = ^^^f{D-2r), and so on; wherefore the value oif{D) will be represented by the infinite product ? !^^, T L", t /"!''/'' • Hence (11) becomes v+f{I)y^v=V 27—2 420 ' FINITE SOLUTION OF DIFFERENTIAL [CH. XVII. with the relations u-P^^v U=P^^V (12) As this proposition is of great importance in the solution of differential equations, it will be proper to examine the condi- tions which its application involves. Evidently they con- sist in such a choice of the form of -v/r [D) as will render the symbolical product P^ , ) jy. finite, and the transformed equa- tion (11) integrable. That the expression of P^ 7777: "^^7 ^^ finite, it is sufficient that for every elementary factor %(i^) occurring in the nume- rator there should correspond a similar factor ^ (i) + ir) in the denominator, i being any integer or 0 ; and vice versa ; for ^ % (^ + /r) xi^ + ^'^') % {^ + (^* - 1) ^'}- • • 1 X{I) + ir)x{D+{i-l)r]...x{n-\-7y wliicli is a finite expression. Again ■^' X[r>-ir) ~x{D- ir) x{D- (» + 1) r}... ^x{D)xiD-T)...x[D-{i-\)r], which is also finite ; the product of any number of such ex- pressions is finite also. Hence, if %(X>) be any elementary factor of <^ (D), it may be converted into % (i) + ir) ; for let ^ [D) = ^ {D) ^^ {D), and let '^[D) = x[D ± ir) xX^)^ wherein %i(-D) denotes the pro- duct of the remaining factors, then p4>iD)_-p x(^) 't(^J "xi^t"-)' Tvliicli is finite. AET. 4.] EQUATIONS IN THE SYMBOLICAL FORM. 421 Hence also, if (p [D) involve any factor of the form —^ it may be made to disappear ; for let ^ (Z>) = ^J^ . . n^^ (Z>), and let f {D) = Xi {D) , then which is finite. 4. We see, then, that there are two distinct kinds of trans- formation to which the Proposition may be applied. In the first kind (/> [D) is converted into another symbolic function •^ [D) without any loss of component factors, whetlier of nu- merator or of denominator, but only Avith such change as consists in the conversion of D into D ± ir. And here the order of tlie transformed equation is the same as that of the equation given, and, its solution introducing a sufficient num- ber of arbitrary constants, no others need to be introduced, either in the prior determination of V or in the subsequent derivation of u. But in the second species of transformation some component factor of (/> [D) (usually of the form y: — y where a — h is a multiple of r) is lost, and the transformed equation being of an order lower than that of the equation given, the deficient constants of its solution must be supplied, either beforehand in the determination of V, or subsequently in the derivation of u. If in the former, any constants, sufficient in number, introduced by the performance oi ' U will serve the purpose. If in the latter, all 1 't(^). the constants introduced by the performance of P^ pry-r must be retained, but their subsequent relation must be de- termined by means of the differential equation. The reason why the constants connected with the disap- pearing factors are arbitrary in V alone, is, that V enters into no other equation than the one in whose solution those con- stants are found. If, however, the entire series of constants in V be retained, they will be reduced by the subsequent differentiations in passing to the value of u. 422 FINITE SOLUTION OF DIFFERENTIAL [CH. XVII. All tliat may seem obscure in the above statement will be made clear by the following examples. Ex. 6. Given -j-.^ + q~u ^ = 0, an equation occurring in the theory of the earth's figure. The symbolical form is Kow we may, by Prop, iii, directly reduce this equation to the form which, by Prop, i, is resolvable into two equations of the first order. But it is better to assume as the transformed equation '^171^/'^ = ' (^)' the solution of which is known already. Art. 2. By Prop. II, assuming u = e~^^ w, we have Again, by Prop. Ill, we can pass from (c) to {h) by assuming Hence u = e''^ {D -1) {D - S)'v on restoring x and putting for v its value in terms of x. ART. 5.] EQUATIONS IN THE SYMBOLICAL FORM. 423 Effecting tlie difFerentiations, we find ^ " "^ i(^~^ / ^^^ ^^"^ "^ "^'^ ~ "J" ^°^ ^^^ "^ ^'^1 ^'^' We might have proceeded directly from (a) to (Z/) by Prop. Ill ; but, had we done so, the final reductions would not have depended on differentiations alone. Thus we sliould have had p D{D-\) _D-l '' --^^ (D + 2) {D - S)^~ n -{- 2^ = (I _ 3 (1> + 2)-^} v = {l- Se-'^I)-V^)v whence, restoring x and giving to v its previous value, we should be led to the same solution as before. 5. The two forms of solution above presented illustrate an important observation, viz. that when in the transition from -.-M)-"-^- ^"^- (J \ — " -— J A", i. e. the result 424 FINITE SOLUTION OF DIFFERENTIAL [CH. XVII. obtained by writing e^ for x in the ?i*'^ integral of Xcix"^^ no constants being added in the integration. From inspection of (a), it is evident that the class of equations sought must, on assuming x — e^, be expressible in the form [D + aj (Z) + a^) ... (i> + a,.) in which we shall suppose the quantities a^, a^..,an to be ranged in the order of decreasing magnitude. Put u = e~''^^u^^ then by Prop, ii, u,±-rrrrrr v^TTTT .e"\ = e"^^Z7 (c). The first factor of the denominator of (j) [D) in (c) now agrees with the first factor in that of 'y\r[D) in (a). In any of the remaining factors we may, by Prop, ill, convert D into D ± in, I being any integer, — hence, that they may all cor- respond with the factors of -^/^ {D) , it is necessary that each of the quantities a^ — a^ + 1 ^3 — ttj + 2 ^4 — cii + S a„ — a^ + n — l . ^^ n n 71 n ^ should be equal to a negative integer or to 0. And in this statement the conditions of finite solution are involved. The value of u will be deduced from that of v by differenti- ation, for since a^ — a^< — l, P' (/> + ~- a,) -{D-l){D-n + \)...{D + a,-a^ + n), and so on for the remaining factors to Avhich P,i is to be applied. Lx. 8. Given -^ , u ± qhc = 0, where ^ is an CCX X integer. This equation, which includes that of Ex. 6, presents itself in various physical problems (Poisson, Theorie Mathematique de la Chalexir, p. 158. Mossotti, on Molecular Action, c&c). ART. 5.] EQUATIONS IN THE SYMBOLICAL FORM. 425 Its symbolical form is Hence, by the last example, = e-^(Z)-l)(D-3)...(Z)-2^ + l)^' (J), where v is given by -j-^ ± (fv — 0. The expression (J) may be reduced to a more convenient form, as follows. Since f{J) -a) = e"^f{I)) e-«^ we have ~ ' \ dO) ^ "" Hence, according as the upper or lower sign is taken in the original equation, we have 1 / ^ d\- c, cos qx + c„ sin qx . . « = ^'(^ SJ of» ('=)' "=^r^j a?-^ ; ('^)- Ex. 9. Given p, -a'p,- i^i+iil' = 0. ctx ciy X Comparing this equation with the last, we see that its solu- tion may be derived from {d) by changing therein q into a—, and c^, c^ into arbitrary functions of y following the exponentials. Hence we shall have 426 FINITE SOLUTION OF DIFFERENTIAL [CH. XVII. ""'^^^ V dx) W __ 1 / 3 dV (l>{7/ -\- ax) + -v/^ (.y — ax) X The reason wliy the arbitrary function (j){y) must be placed after e^'^d'y and not before it, is that, in the derivation of the exemplar form, the arbitrary constant takes its place after, and not before e^"". For g-,)-'o = e''(0'o = e''o. Here indeed we may transpose the constant, but when q is converted into a -y- we have and here the arbitrary function cannot be transposed, since y , d and -r- are not commutative. ay The principle here illustrated, and which is a very important one, is that all conclusions founded on community of formal laws should stop short of interpretation. Thefoivn should be kept distinct from the matter. There is perfect analogy between the theorems but not between the theorems because in the formation of the latter interpretation has been employed. ART. 5.] EQUATION^ IN THE SYMBOLICAL FORM. 427 The above example is one in which Monge's method of solution would fail, except for the particular case of 1 = 0. And this gives occasion to the remark that symbolical methods are not, as they have sometimes been supposed to be, valuable only as abbreviating the processes of analysis. There are in- numerable cases in which they afford the only proper mode of procedure. Ex. 10. Given This equation occurs in some researches of Poisson on definite integrals. The symbolical form is This equation is integrable in several distinct cases, but we shall examine here the particular case in which n is an integer. Assuming as the transformed equation, it being necessary to introduce V because the transformed equation is of an order lower than that of the equation given, we have, w = P, ^ V = {i) + 2n-2) (D + 271-4)... (i)+2)v (c), 0 = (i) + 2?i-2)(Z> + 2/i-4)...(Z) + 2)F. The latter equation gives for V the general value, of whicb it suffices to retain one term. Eetaining the first, 428 FINITE SOLUTION OF DIFFERENTIAL [CH. XVII. substituting in [h), and operating on both sides with D+j^^ we get (!) + p)v-{D + 2n - 2 -;p) e~^v = c,(p- 2) e'^^. Eestoring x, and integrating, a value of v is found, involving two arbitrary constants, whence u will be given by « = ("i + 2'^-2)(^l + 2»-^)-(-£ + 2).....(cZ). The proposed equation is also integrable when^j> is an odd integer, and when 2;i— ;pis an even integer. In the former case we maj assume as the transformed equation, (I) + 2n-l-p)iD + 2n-l-p-l) {D + p)[D+p-l) ' "-"' which must be integrated by Prop. i. In the latter case we must assume V — e^^v = V; but in this case two constants must be retained in V; viz. one from each set of the reducing operations by which the factors of (p [D) are made to disappear. 6. It will be observed that, in the foregoing examples, we reduce the proposed symbolical equation by Propositions II. and III, either directly to an equation of the first order, or to a form which by Prop. I. is resolvable into a system of equations of the first order. But there exist other equations admitting of finite solution ; for example such as by Props. Ii. and III. are reducible to either of the primary forms, -^^f^iS^-"" ("). -'^^iSJH^--" M- The former of these is the symbolical form of the equation ,, ov d% du „ ^ (1 + ax) -j-7. + cix-r- -^ nu = 0, ART. 6.] EQUATIONS IN THE SY^IBOLICAL FOKM. 429 wliicli Is reducible to -7-7 ± n^^ = 0, by the assumption The latter is the symbolical form of the equation {x' 4- a) x' g + {2x'-\- a)x^± nhi = 0, which is reducible to -^ ± ?z-ii = 0, by the assumption J x\/(x^ + a) ^(x' + a) Hence, the ordinary solutions of (13) and (14) will be obtained by substituting t^l—^ t={ ^^ j ^{l + ax')' J x^/{x' + a) ' dhc in the solution of the equation -y^ ± ^^^^ = ^' It maybe added that the forms (13) and (1-1) are allied, the one being convertible into the other by changing ^ to — ^. Ex. 11. Given The symbolical form is '' l)[D-\) ' "-^• If we apply Prop. 11. so as to convert D into D — m, and then by Prop. iii. reduce the equation to the general form (13), we shall obtain the final solution in the form Jl\ w = f -y- j {^1 COS {([ sin~^ x) + (?2 sin (// sin ^ x)]. 430 FINITE SOLUTION OF DIFFERENTIAL [CH. XVII. 7. Pfaff's Equation. The differential equation, {a + hx-)x'^, + {c-tex^)x^+{f+gx'')u = X (a), wliicli includes all binomial equations of the second order, has been discussed by Euler, and, with greater generality, by Pfaff [Disquisitlones Analyticce). We propose to investigate the conditions under which it admits of finite solution. It suffices for this purpose to consider the case in which x=o. • The symbolical form is then ^ + aD[D-l) + clJ+f ' u-0,...{b]. If 71 is not equal to 2, it is convenient to change the inde- pendent variable by assuming nd = 20', whence d _n d de'~2dd" 71 So that changing n6 into 20', we must change Z> into - 1), The result may be expressed in the form, where a^ and a„ are roots of the equation, lQ-n){"^-n-l) + e[^-n)+, = 0 (d), and /Sj, ySa ^^^ ^^o^s of the equation, <^'^{f-^)^4^f=' (^)- 1st, By Prop. Ill, (c) can be immediately reduced to the form MJ-..)(.P-a.-l) '' + a(i>-^J(i>-^.-l)' ART. 7.] EQUATIONS IN THE SYMBOLICAL FORM. 431 and then resolved into two equations of the first order, if we have at the same time a^ — a^, and ^^ — yS^ odd integers. 2ndly, The equation can, by Prop, iii, be reduced to an equation of the first order if any one of the four quantities ^i-A) «i-A2' 0^2-^' Wa-ft is an even integer. 3rdly, It is easily shewn that by Props. 1 1, and ill. (c) is reducible to the integrable form (13) if the quantities y8j-A and a^ + a^- ^,- /S^ are both odd integers. 4thly, It is in like manner reducible to (14) if the quanti- ties flfj-ofa and cc^ + a^- 13^-/3.2 are both odd integers. These results may be collected into the following theorem. The equation (c) is integrable in finite terms, 1st, if any one of the four quantities represented by a — /3 is an even integer ; 2ndlyj if any two of the quantities are odd integers. In this theorem the integral values are supposed to be either positive or negative, and the even ones to include the value 0, The above results are equivalent to those of Pfaff, as pre- sented with some slight increase of generality in a memoir by Sauer {Crelle, Vol. ii. p. 93). PfafF's conditions are how- ever exhibited in so complex a form as to render the com- parison difficult. His method, it is needless to say, is wholly different from the above. 432 SYMBOLICAL EQUATIONS [CH. XVII. Symholical equations loMcli are not hinomial. 8. Althougli processes of greater or less generality may be established for the treatment of equations which, when symbolically expressed, involve more than two terms in the first member, yet their reduction if possible by some preli- minary transformation to the binomial form should always be our first object. We purpose here to illustrate this observa- tion. Ex.12. Given ^= a ^^ Writing this equation in the form (2c -xYx''^,-{-hy==a {2cx - x')\ we see at once that its symbolical form will not be binomiaL Assuming y— {2c—xy\i, we have on reduction ^ ^ aic ax 2c — X (2c — x) Now let on be so determined as to make the numerator of the third term divisible by its denominator. This involves the condition m{m-l)+-^, = 0 (a), while the differential equation becomes cl^ii c7ii ax? (2c -x)x^^,- 2mx'' ^ - m {m - 1) (2c -\-x)u= ^^^_^y.-i , of which the symbolical form is {D-m) {D+m-l)u- -- [D+m-l) {D+m-2)e^u= ^ ^ 2c' ' '' ' ^ 2c{2c-eY whence, operating on both sides with {D + m — 1)~\ {D-m)u-l- [D+m - 2) e^u = ^ e^^-'^^f D"^ e"-''^'^^ (2c - e^}' JiC zc ART. 8.] WHICH ARE NOT BINOMIAL. 4o3 Kestoring x, and solving the equation, we have, on representing 2c — ic by X, u = ax'^X'-^'^' ix-'^'X''''-^ jx'''X'-'^dx\ which integration by parts reduces to the form a ix'^X'-'''' jx^'-x'-^'dx-x'-"' L'^^X'-'^'dx ^"^ 2c(2m-l) "' a L'^X^-" jx'-'^X'^Ix -x'-''X"'lx'\X'-''dx Therefore ?/ = — ^ — —r -? , "^ 2c {'2m - 1) the integral required. It is to be noted that each integration introduces an arbitrary constant. It is also seen that each value of m derived from (a) leads to the same result. The above equation occurs in the problem of determining the tendency of an elastic bridge to break, when a heavy body, e. g. a railway train, passes rapidly over it. The equation between y and x is, on a certain hypothesis, that of the tra- jectory described. See an interesting paper by Prof. Stokes [CambiHdge Phil. Transactions, VoL VIII. p. 708). Ex. 13. Given (1 - ix') ^{l- fi') j +n{n + 1) (1 - fj}) u d% ^ Eepresenting ti \/(— 1) ^7 «> ^^i® equation may be ex- the well-known equation of Laplace's functions. Eepresenting -7-7 pressed in the form (1 - H-r J - 2/. (1 - f^') J + {« (« +1) (1 - /.') - «^« « = 0, and it is evident that it would not, on assuming /x = e^, take the binomial form. Let then u={l~fiyv. We find, on substitution, and division of the result by (1 — fxy^\ B.D.E. 28 434 SYMBOLICAL EQUATIONS [CH. XVII. Let 4:r^ — a^=0. Then r=±-. Either sign may be taken. Choosing the lower, we have (1 -/.^) ^ + 2 (a - 1) /.|^ + {n [n + 1) - a (a - 1)}^; = 0, an equation which, on making /x = e^, assumes the symbolical form _ [D~-a + n-l){D-a-n-2) ^ ^ D{D-l) ^ ^ ^ ^^^' To integrate this, assume Then by Prop, iii., v = {D-a + n-l){p-a + n-2>)..,[D-a-n + l)w ='^"ili)>- ('^)' while (c), resolved by Prop. I. and integrated, gives the solution w = (1 + ^r^^ {4>) + (1 - /.)""x () («). yfr and x being arbitrary functional signs. This expression for w having been substituted in (d),WQ must write TrV'(— 1) for a, and interpret the result. Now if, instead of '^(0) and %(^), we write i^{6'^V(-i)} and ^Je0V(-i)j^ as we are evidently permitted to do, and if we observe that generally ART. 8.] WHICH ARE NOT BINOMIAL. 435 -4 m we shall ultimately find where Fl,, e*V.-«| = ^" (A l)"{(^+ ^y^(^') + (/^-mT%(^^j} a^)' wliicli is the complete integral. For a discussion of this result, and for the finite expression for Laplace's functions to which it leads, the reader is referred to a paper on the Equation of Laplace's functions in the Camhridge Mathematical Journal. (New Series, Vol. I. p. 10.) If in the equation (a) we make the third instead of the fourth term to vanish, which gives for r the values - and — - , and then assume — — ^- — ^r- = t. we shall obtain, 2 V(l-/^) taking the second value of r, the symbolical equation (D + n-lf-a' ,„ ^ "+ D[D-l) ''' = '' Now by Propositions Ii. and ill. this is reducible to the inte- grable form •-%§^-"-'. by the relation 2S— 2 43G SYMBOLICAL EQUATIONS NOT BINOMIAL. [CH. XVII. \dtj Hence we find «=(|)"\(<+v(i+or+c.!<+v{i+on. whence u is known. Let us examine the form of the solution, when, as is com- mon in tlie expression of Laplace's equation, we replace /m by cos 6. We find ^ = cot ^, -t:= — sin^^ -j-a > at ciu whence t + sJ[l-\-e)= cot i 6, Substituting, and observing that u= (sin^)"''~^i;;we have u = (sin 6)-'-^ (sin'^^)"" [o. (cot f)% c, (tan |)] . And hence, restoring to a its meaning, introducing arbitrary functions for constants, and effecting one of the differentia- tions, we may deduce the following solution of Laplace's equation, viz. : u = (sin 6)-'' (sin 6 ~ sin Of F^ |e^V(-i) tan ^l '"^ ...(16). ..{ n e-«^V(-i)tan- Under this singidarly elegant form the solution, obtained by a different method, was given by Professor Donkin. {Philo- sophical T^^ansactionSy for 1857.) ART. 9.] SOLUTION OF LINEAR EQUATIONS BY SERIES. 437 Solution of linear equations hy series. 9. Prop. IV. If a linear differential equation whose second memher is 0 he reduced to the symbolical form f,{D) u +f,(D) e'u+f,{D) e^'u... +/„(Z)) e"";. = 0 ... (17) (Art. 1), then a particidar solution will he u=Xu^e-' (18), the value of the index m in the first term heing any root of the equation f {m) = 0, the corresponding value of u„^ an arbitrary constant, and the law of the succeeding constants heing expressed hy the equation^ f H u^ +f iin) u„^^ +f (m) u,^^ ...+/„ {m) w,^„ = 0. . . (19). For the form of u assigned in (18) will constitute a solution of (17) if, on substituting that form for u in the first member of (17) and arranging the result in ascending powers of e^, each coefficient should vanish. And this, as we shall see, will take place if the coefficients are subject to the relation expressed by (19). Assuming then u = Sw^je"'^, we find, f,(D) u = tf,{D) K.^e"^ = 2/„(m) u^e"^, by (2), and so on. In the first of these, we see that the coefficient of any particular term e""^ is f{m)u,,,. In the second, the co- efficient of e^"'"'^'^ is f{m + l)u„,, and therefore the coefficient of €*"^ is /i(*^0*^»n-i' Ii^ the third, the coefficient of e'"^ is fi ("^) ^'m-2 > ^^^ so on. Thus the aggregate coefficient of e"*^ is /o H «^m +/, (m) u,^^ +f [m) u„^^^ ...+f {m) «„^„ , and this, equated to 0, expresses the law (19). 438 SOLUTION OF LINEAR [CH. XYII. Let ii^e^'B be the first term in the developed value of u ; then must we suppose u^_^ = 0, u^_^ = 0, &c. and (19) becomes As, by hypothesis, Uy is not equal to 0, this gives f^ (r) = 0, for the determination of r, and leaves w^ arbitrary. Hence the proposition is established. Thus there will, except in particular cases of failure here- after to be considered, be as many distinct solutions of the form (18), each involving an arbitrary constant, as there are units in the degree of /^ {m) . ^ ^ ^ ^ . d^u a — ldu „ JliX. 14. (jTiven -j-r, = nu = 0. ax X ax The symbolical form is Hence, we have u = ^u^x"^, the law of formation of the coefficients being m [m -a)u^- n^u^^ = 0, or u^ = — — -r u^_^ , while the initial exponent is 0 or a. There are therefore two ascending series, one beginning with (7, the other with Cic". Thus we have 2 (2 — a) 2 . 4 (2 — a) (4 — a) ■^^^ +2(a4-2)'^2.4.(a + 4)(a+2)+^'^- 10. When the equation f^ (m) = 0, has equal or imaginary roots, the following procedure must be adopted. Let the solution of the equation j^ (^) w = 0, be u = AP+BQ+CR + &Q (20), A^ By Cj &c. being the arbitrary constants. Substitute this ART. 10.] EQUATIONS BY SERIES. 439 value in the given differential equation, regarding A, B, C, &c. as variable, and the result will assume the form A'F+B'Q+C'E + iSic, = 0 (21), and will be satisfied if we have A' = 0, B' = 0, O'=0, &c (22). This will indeed become a system of linear simultaneous equations for determining A, B, C, &c. And the solution of this system in a series will be of tlie form the law of formation of the coefficients a^, h^, c^, &c. being expressed by a system of simultaneous equations formed from (22), by changing therein every term of the form {D) e'^ A into (j) {in) a„j_i, &c. {Philosophical Transactions.) There is a particular case of exception to the above rule. When two of the roots of /^ [m) = 0 differ by a multiple of the common difference of the indices of the ascending develop- ment, the equation y^ {D) = 0, must be replaced by what that equation would become were the roots in question equal. -n . ^ ^ . ^"^ 1 du „ Ex. 15. Given -7-^ + - ^r- + qu = 0. The symbolical form is • D\i + q'e'^ar=0 (a). Now B^u = 0 gives u = A + BO. Substituting this value in (a), regarding A and B as variable, we have D'A + q\'' A + 2Z>i? + 0 {B'B + q'e'B) = 0, which furnishes the two equations, B'A + q'e'' A + 2DB = 0, D'B + q'e'^ B=0, whence A = ta„, e"'^ B = tb,^ e"'^ with the relations m\, + q'a^_^ + 2mh^ = 0, m'b^ + q^h,,^^ = 0, SOLUTION OF LINEAR [CH. XVII, wliich we have 7. -?\ -^ . 2/7. w. Thus we find, on substitution, and restoration of x, u = a^-{- ajoc^ 4- ct^x^ + &c. + log X (5, + hx"" + l^x" + &c.) , where a^^ h^ are arbitrary, and the succeeding values deter- mined bj {b). Were the symbolical equation of the form it would still be necessary to determine the form of the primary assumption by solving the equation D^'u = 0, not by D {D± 2i) u = 0. We should therefore still have u = A+ Bd, in which A and B are series to be determined as before. Ex. 16. Given x'^. + x'^-V in' + a;> = 0. ax ax The symbolical equation is {D' + 7i')u + 6'hc = 0 («). Now the equation (D^ + n^) u = 0 gives w=^cos w^ + ^sin w^ (5), substituting which in (a), and equating to 0 the coefficients of cos 716 and sin nd in the result, we have B^A-\-2nDB + €'^A = 0, I)'B-2nDA + €'^B=0, whence A — Sa,«6*"^, B= Xh^^e""'^, with the relations, m^a^ + 2mnh^ + «,^ = 0, m'5^ - 2m?2«,„ + ^«_2 = 0, ART. 11.] EQUATIONS BY SERIES. 441 and therefore, ^"» m{m'-\-4.n') ' "^"^ 'm {m' + An') ^^^' Thus the solution assumes the form, u = cos {n log cc) {a^ + ag^?'^ + «4^* + &c.) + sin (w log x) {h^ + K^x" + h^x^ + &c.), wherein a^ and &,, are arbitrary, and the succeeding coefficients determined by (c). The fundamental equation (19), written in a reversed order, determines the law of the formation of the coefficients in those solutions of (17) which are expressible in descending powers of x. The number of such solutions will be equal to the degree of the equation /„ [m) — 0, but their respective first exponents will be its roots severally diminished by n. For the extension of the above theory to the case in which the given differential equation has a second member X, the reader is referred to the original memoir. Theory of Series. 11. The relations which enable us to express the integrals of differential equations in series, enable us also to reduce the summation of series to the solution of differential equations. Thus, from Proposition iv. it appears that if w = ^u^x''\ where the law of formation of the successive coefficients, is /„(m)w.^+/,(m)t._,...+/.(m)w_ = 0 (23), the value of u will be obtained by the solution of the differ- ential equation, /o (^) ^^+/ {D) e^c ... +/„ (D) rt. = 0 (24). We suppose here /, (m),f (w).../, {m), to be polynomials, and that the series is complete; i.e. contains all the terms which can be formed in subjection to its law expressed by (23), the first exponent being therefore a root of /^ (w) = 0. "Wm / - X ^^jrt-2* 442 THEORY OF SERIES. [CH. XVII. Y/hen the series is incomplete, the first member of the differ- ential equation will be the same as for the complete series, while the second member will be formed by substituting in the first member, in the place of u, the series which it repre- sents. It is obvious that all the terms will disappear, except a few derived from that end of the series where the defect of completeness exists, so that the second member of the differen- tial equation will be finite. Ex. 17. Let u = l x^ H ^ -x^ ^ — ' x\ &c. 1.2 1.2.3.4 1.2.3.4.5.6 ' Here zi = ^u^x^\ with the relation, n^ - (771 - 2y m (m — 1) Or, m (m- 1) u^- {(m-2y -n'} u^, = 0, and we observe that the series is complete, the first index 0 being a root of ?m (wz - 1) = 0. Hence, the differential equation will be I)(D-l)u- {{I)-2y-7i'} i^u = 0, of which the solution, expressed in terms of a?, is w = Cj cos [n sin~^a?) + c^ sin {n sin~^ x). The constants must be determined by comparison with the original series. We thus find c^ = 1, c^ = 0. The following is a species of application which is of frequent use in the theory of probabilities. Ex. 18. The series ART. 11.] THEORY OF SERIES. 443 occurs as the expression of the probability that an event whose probability of occurrence in a single trial is ^, and of failure 5', will occur at least a times in a + 6 trial^. Representing the series within the brackets by w, and assuming ^ = e^, we liave u = Sw^e"*^, where mu^ — (m + a - 1) u^^ = 0. Hence, we shall have 1 .2 ... 0 or, restoring ^, du a _ a (a + 1) ...(« + &) ^ dq 1-q 1.2...& 1-q Integrating which, we have .^(i-gr{c-"("+'^--/;+^)£g>(i-grvgi. Now the first term of the development of this expression in ascending powers of 5' will be C; whence, comparing with the bracketed series, we have (7=1. Substituting, and observing that p = '^ — q, the expression for the probability in question becomes _ a{a + l) ... {a + h) 1.2. ..6 To this Ave may however give a more symmetrical form. For /' a' (1 - i)'-'dq = (£ - £) q' (1 - qr\dq by a known theorem of definite integration. Substituting in (a) , and observing that a[a + l) ... {a-\-h) _ T (a + h + 1)_ 1.2. ..6 ~T{h-{-l)T{ay [\'{l-qrdq {a] J 0 444 THEORY OF SERIES. [CH. XVII. we find or, as it maj be otherwise expressed, r»"%=i¥{i^|S| «. The peculiar advantage of this form of expression is that, precisely in those cases in which the series becomes unmanage- able from the largeness of a and h^ the integrals admit, as Laplace has shewn, of a rapid approximation [Tlieorie Ana- lytique des Prohahilites). Ex. 19. The function (1 — 2v cos w + O'*" heing expanded in a series of the form ^^, + 2 (^.^cos o) + ^2C0S 2a) ... +&c.), it is required to determine A^. We have (1 - 2v cos (o + v')-'' = {1 - ve^'^'^-'X" X {1 - ve-'^^^-'Y- Expanding each factor, and seeking the common coefficient of g»-a,v(-i) and e~^"^^^"^^ in the product, we find, putting t = v^, where generally, m (m ■^T)u^-(7ii^-n- 1) [m -\-n-\-r-l) w„^, = 0, ^^^^^" ^^= -^ — r:2Tr^^ • Hence the differential equation will be, Z) (D + r) w - (Z) + n - 1) (D + w + r - 1) e^w = 0, D{I) + r) Now this can, by Prop, iii, be reduced to the fonn, v-e%= F, ART. 12.] THEORY OF SERIES. 445 by the relations, u= {D-Vn-l)...{D + l)[D-\-n + r-l)...{D + r + l)v, F={(i) + n-l)... {D -rl) [D + n + r -I) ... [D + r + l)Y'U, In determining Ffrom the latter equation, it suffices to in- troduce two arbitrary constants, one from each of the two sets of inverse operations. The final sokition, in the obtaining of which the only difficulty consists in the reductions, is 12. When, in the series S^^,,,^"*, the coefficient u^ is a ra- tional function of m invariable in form, the summation is most readily effected in the following manner. Let the series be 2(/) (m) x"'; then putting x = e^, = 0(i))26'"^ (25). Hence, if the summation is from 7n = 0 to 77i = infinity, we have but if the summation is from m = a to ??i = h inclusive, u = (t>{D) -n «^ T . 4a;^ 5x* Qx^ p Ex. 20. Let „ = ^-^^ + ^-^ + 3-^, + &c. Here <^ {m) = 111 [m — l) {in — 2) ' ii)->-2(i)-ir+-5(i?-2r|-^ 446 GENERALIZATION OF THE [CH. XVII. The final result is Generalization of the foregoing theory, 13. As Propositions I, ii, iii, are founded solely on the particular law of combination of the symbols D and e^, ex- pressed by the equation they remain true for any symbols ir and /j, whatever their interpretation, which combine according to the same formal law; viz. f{7r)p'^u=py{7r + m)u (26). Thus, supposing the law obeyed, the symbolical equation, w + 0(7r)/3"w= U (27), can, by Prop. ill. considered in its purely formal character, be transformed into V + yfr {tt) p^'v = V (28), by the assumption, Y (tt) Y (tt) The corresponding transformations flowing from Proposi- tions I and II, it is unnecessary to state. Now the law (26) is obeyed, not alone by the pure symbols D and e^, but by certain combinations of those symbols. Thus, if we assume '7r = D-n(f>{D)e', p = [D)e' the law will still be obeyed. And the importance of the remark consists in this, that an equation which, when ex- pressed by means of the symbols D and e^, is not a binomial, may assume the binomial form for some other determination of TT and .0. ART. 13.] FOREGOING THEORY. 447 If in (26), we make m=l, wehsiYe f{7r)pu = pf('7r+l)u, which shews that p may be transferred from the right to the left of /(tt), if we, so to speak, add to tt the constant incre- ment 1. This then suggests the more general law, f{7r)pu=pf{7r+A7r)ic (29), where Att represents any constant quantity regarded as an increment of tt. In connexion with tiiis theory, the following proposition is important. Prop. Supposing f (x) to represent a function icliich admits of expansion in ascending positive and integral powers of x, it is required to dev elope f {tt -\- p) in ascending powers of p^ IT and p being symbols which combine in subjection to the law (29). By successive applications of (29) we have, m being a positive integer, f{'n■)p''\^,^py{7^ + m^'7^)u (30), of which another form is p^fi^r) u =f{7r — iiiAtt) p^u. Again, since y(7r + p) is, by hypothesis, expressible in a series of the form A^ + A^(tt -t p) + A,^{7r + pY i- &c. we shall have {'jT i- p)f{7r + p)=f{7r + p) {ir-h p) (31), for either member becomes, on substituting fov fijr + p) the above form, A (tt + /)) + A^ {it + pY + &c. Now, let the form of the unknown and sought expansion of / (tt + p) in ascending powers of p, be /{■^ + P) =/o (t) +/. (tt) p +/, (tt) p= + &c (32), the Ksubject u being understood tliougli not expressed. Then, b7 (31), (tt + p) S/,. (tt) p" = S/,. (tt) p" (,r + p) . m+1 i 448 GENERALIZATION OF THE [CH. XVII But (tt + p) tf^ (tt) p- = tirf^ {tt) p- + tpf^ (tt) p- = S7r4(7r)p-+:£/;.(7r-A7r)p"- in which the coefficient of p"^ is ^/.W+/«.-x(^-A7r) (33). Again, SX (tt) p- (tt + p) = ^y;. (tt) p-TT + 2/^ (tt) p-^' = S^ (tt) (tt - m Att) p- + ^/,, (tt) p in which the aggregate coefficient of p"' is fm W (tt- wiA-Tr) +/„_, (tt). Equating this with (33), we have 'rrfm i'rr) +fm-^ (tt - Att) = (tt - mAtt)/, (tt) +/,_, (tt), whence •^- (^) -m A^^ if we define A/(7r), not, as is usual, 'by/(7r+ Att) -/(tt), but hy /(t^) — /(tt" — Att). The above equation determines the law' of derivation of the coefficients /^ (tt), f^ (tt), &c. It only remains to determine /„ (tt). That /„(7r)=/(7r) may be shewn by induction from the particular cases in which /(7r + p)=7r4-p, (7r + p)^&c. or, with more formal propriety, thus : Let Pj = np, where w is a constant, / W Pi =/ W ^P = ^^M P = npfiir - Att) ART. 13.] FOREGOING THEORY. 449 Comparing the first and last members, we see that ir and p^ combine according to the same law as it and p. Thus, we have, /(^ + P.) =/o W +/i (^) Pi +/. M P' + &C. foMi fi My ^^* being the same as in (32). Or, /{■n- + np) =/„ (tt) +/. (^) n/> +/, (,r) nV + &c. ; so that, making w = 0, we have^ (tt) =/(7r). Determining then the successive coefficients by (34), we have finally, N r>; s ^fM 1 AY(7r) . • ^ ^^(-)p^4-&c (35), 1.2.3 (Att) wherein it is to be remembered, that A/(7r)^/(7r)-/(7r-A7r) Att Att When Att = 0, the symbols tt and p become commutative, and (35) assumes the form of Taylor's theorem. As a particular application of the above, suppose that we have given the trinomial equation ^])'+aD + h)u+(cD + e) ehc + fe'^ a = 0 {a), and that we desire to ascertain whether this can be trans- formed into a binomial equation by assuming TT = D — 7we^, p = 6^, assumptions which satisfy the law /{■7r)p = pf{ir+}). Here we have i) = tt + mp, whence f{D) =/(^) + ^ ,np + ] ^g^ my + &c., E.D.E. 29 450 Laplace's transformation of [ch. xvii. where A7r = l, and Hence i)^ + aD + & = tt'^ + ^tt + Z> + (27r - 1 + a) mp + m'p' , cD + e = CTT + e -f cnip. Thus (a) becomes {tt^ + a-TT + & + (27r -l+a)mp + m^p^} u + (CTT + e + cmp) pu +fp^u = 0, or 7r'+a7r+& + {(2m+c)7r + m(a-l)+e}p + (?^'+cm+/)p'=0, and this reduces to a binomial equation, 1st, if ??i be a root of the quadratic equation 2ndlj5 if it be possible to satisfy simultaneously the equations 2m + c = 0, m{a- l) + e = Oj equations which imply the condition 26 - c (a - 1) = 0. The discussion of the binomial equation when obtained in- volves no difficulty. For a discussion of the general trinomial equation of the second degree, the reader is referred to the original Memoir. Laplace's transformation of partial differential equations, 14. Laplace has developed a method for the reduction of the partial differential equation Br+Ss+Tt + Fp+ Qq+Zz= U (36), B, 8, T,...Z7 being functions of x and y, which is deserving of attention from its great generality. One of the auxiliary equations in Monge's method is Jld2^^ _ Sdxdj/ + Tdx' = 0. ART. 14.] PARTIAL DIFFERENTIAL EQUATIONS. 451 Let two integrals of this equation be {x, y) = C, yjr (x, y) = c', and assume two new variables, f and 77, connected with x and y by the equations i-={x,y), 'n = ^\r{x,y). The student will have no difficulty in proving that the given equation will assume the form Z, il/, N, F being functions of f and 77. The theory of the reduction of this equation is then contained in the following propositions : 1st, The equation (37) may be presented in the form Hence, if the condition N-L3£-^ = 0 (39) be satisfied, and we assume ly- + L] z = z', we shall have F. 01-^ ^y The solution of the given equation is then dependent on that of two partial differential equations of the first order. 2ndly, Inverting the order of the symbolic factors, the equation is also solvable if we have N-LM--^- = 0 (40). arj 3rdly, The equation (37) can be transformed into a series of other equations of the same form, and tlierefore integrated, if, for any of those equations, the condition (39) or (40) is satisfied. 29-2 452 LAPLACE'S TRANSFORMATION OF [CH. XVII. For, expressing it in the form (38), let, as before, ii^^)-' (")• Then (J- + II\ z' + (N- LM- ^) z = V, whence z = which is of the form d^ A, B, C "being functions of f and 77. Substituting this ex- pression for z in (41), we have a result of the form ^''^' dz dz ^L'^-\-M'^-\-N'z'=r (42). d^ dr\ c?f dt] Thus the form (37) is reproduced, but with changed coeffi- cients. Hence the equation is integrable if either of the fol- lowing conditions is satisfied, viz. N'-L'M'-^ = 0, N'-L'M'-^=0 (43). a^ dr] ^ ^ If neither be satisfied, the process of transformation may be indefinitely repeated, and should an equation be obtained in which either of the relations (43) is satisfied, the solution may be found. It has indeed been asserted that "if the given equation be integrable, we shall finally get an equation in which this essential condition is satisfied" (Peacock's Exam- ples, p. 464). The state of our knowledge of the conditions of finite integration does not however warrant this confidence. A discussion of the equation dz „dz d^z ^ ^ d'^z . d'z ^^dx -^dy qz ^r , s dx' dxdy dy' hx + ky Qix+kyY ^ ' ART. 14.] PARTIAL DIFFERENTIAL EQUATIONS. 453 by Laplace's method is given in Lacroix (Tom. ii. pp. 611 — 614), but it is far too long and too complex to find a place here. The best mode of treating the equation is probably the following. Let s and t be two new variables connected with X and 2/ by the linear relations Jix + k7/ = Sj 7/ + mx = t, of which one is suggested by the form of the given equation, while the other is adopted in order to put us in possession of a disposable constant m. Transforming, and making in the result s = e^, we obtain the symbolical equation {AD{D-l)-vED^-g]z-\-j^[B{D---l)-^F]e'z-\-C^,^^^ in which A = ah^-\- bilk + ck\ B = 2ahn + 5 (A + hm) + 2cA', The equation will be a binomial one, if m be determined so as to make (7=0. We have then am^ + Jwi + c = 0, while the symbolical equation (h) becomes and is integrable if the following condition is satisfied, viz. B-F A-E±^J[{A-Ef-^] — Ti -^. ^^ = an mte^rer or 0. B '2A ^ This condition will be found to include the one to which Laplace's method leads. At the same time it is seen that the equation (h) assumes the binomial form under other conditions than the above; e.g. if we have simultaneously ^ = 0, F=0, from which, by elimination of m, we find / (2aA + hh) - e {fill + 2c/v) = 0. 454 MISCELLANEOUS NOTICES. [CH. XVII. This condition being satisfied, and m determined, tlie sym- bolical equation becomes and is integrable if the two roots of the equation Am (m - 1) + Em + g = 0 difier by an odd integer. There are probably other cases de- pendent on the reduction of Art. (13). In one respect Laplace's transformation possesses a gene- rality superior to that of all others. For its tentative applica- tion fewer restrictions on the coefficients of the given equation are necessary. But, that the application may succeed, other conditions seem to be demanded which render the estimation of the true measure of its generality difficult. And, in parti- cular instances, it is seen that it is less general than the method of the foregoing sections. Miscellaneous Notices. 15. Of special additions to the theory of the solution of differential equations by symbolical methods, the following may be noticed. 1st, Professor Donkin has shewn that, if /(a?) be any function capable of development in powers of x, then whatever may be the interpretations of the symbols tt and p, we have fip-^7rp)u = p-y{^)pu (44). This is evident from the consideration of such cases as the following: {p-^TTpy — p~^7rpp~^7rp = />~V^/?, (p-Vp)-' = p-v-'(p-r'=rv> We are thus enabled to generalize many important theorems. Thus, since |^ +^'(x)lu=^ e"^^^) ~ e'^^'^u, we have /{i + ^'H}— /(i)e-^>. (^5)> {Camhridge Mathematical Journal^ 2nd Series, Vol. V. p. 10.) ART. 15.] MISCELLANEOUS NOTICES. 455 2ndl7, Mr Hargreave, observing that tlie symbols y- and — X are connected by the same laws as x and -j- , (the proof of ctx this will afford an exercise for the student), has remarked that if in any differential equation and its symbolic solution we change x into -y- , and -7- into — x, we shall obtain another form accompanied by its symbolic solution. (Philosophical Transactions for 1848, Part I.) Applying this law of duality to the known solution of the linear differential equation of the first order, it is easy to shew that the equation x<^ {D)u + f [D) u = X has for its symbolic solution, u = {6{D)}-'e^^''^x-'e-^^''^X (46), where ^ (D) = |l|g Ji>, a form which had before been established on other grounds, [Philosophical Magazine, Feb. 1847). Many other illustrations of the same law will be found in the memoir of Mr Hargreave referred to. Srdly, The method by which the development off{7r-\-p) is obtained in Art. 13, leads to other and similar results, of which the following is among the most interesting, viz. the coefficients of the expansion in the second member follow- ing the law of Taylor's theorem, and the function F (x) being equal to e^^"'''' f{x). {Cambridge Mathematical Journal j 1st Series, Vol. iv". p. 214.) The last theorem enables us to integrate at once any equa- tion of the form, 456 MISCELLANEOUS NOTICES. [CH. XVII. where F(x) is a rational and integral function of x. For let an expression always finite under the conditions supposed. Then the given equation assumes the form f{7r)u = X, where tt = aj + -r- , and may be treated "by the method of the last section. Other examples of the expansion of functions whose symbols are non-commutative — some of them admitting of a similar application — will be found in the memoir of Professor Donkin above referred to, and in an interesting memoir by Mr Bron- win [Cambridge MathematicalJournaly Vol. III. p. 36). 4thly, Many important partial differential equations of the second order admit of reduction to the form du dv du dv _ dx dy dy dx ' whence an integral u =f[v) may be deduced. Thus the equation dp dq dg^ dp J^ ^ dp \aq dp J (.Ig, where ^ and ^/r represent any given functions of ^ and 2, may be expressed in the form d[^-x) djyfr-y) d{-x) d {jr - y) ^ ^ ^ dx dy dy dx ' whence (j) — x = F('\jr—y) is a first integral. Mainardi has shewn that nearly all the equations which occur in Monge's Application de V Analyse a la Geometrie^ admit either of the above reduction, or of a purely symbolical mode of solution. [Tortolini, Vol. V. p. 161). 5thly, The Author is indebted to Mr Spottiswoode of Oxford for an interesting communication on the laws of combination of symbols which are at the same time linear with respect EXERCISES. 457 rl fJ to -7- , -J- , &c. and linear with respect to a?, y, &c. The following is one of the results. If, assuming d , d d d a partial differential equation can be presented in the form on the assumption that -7- and -^- operate onl j on the subject ity then it can be expressed in the form F[7r^, 'n;^)u = 0, indepen- dently of such restrictive hypothesis. It might be added, that all such equations are reducible to equations with constant coefficients, by assuming iog(«^+y)4=x', iog(^)*=y. To the above might be added many other special deductions, isolated now, but destined perhaps, at some future time, to be embraced in the unity of a larger theory. EXERCISES. 1 . Integrate x^-j-^ + ^x-^ ^ Vw = 0. 2. Integrate {x' -x')^-{x + Sx') ^| + (1 - x) u = 0. 3. Eiccati's equation is reducible to the form Hence investigate the conditions of integrability. The symbolical form is w + jr-y, e("»+-^* w = 0; and this may either be reduced directly by Prop. III. to a form integrable by Prop, i, or, by assuming {m + 2)d=2d', converted into a particular case of Art. 7 in the Chapter. 458 EXERCISES. 4. The equation -^-^ + - -3- + 5m = 0 is integrable in finite terms if a is an even number. 5. The equation -77^ 4 — -t- = hx'^u is integrable in finite 4 ii + r) terms if m = — ~ , where i is a positive whole number or 0. 6. The more general equation d^u r du /, ^ c tZoj^"^ a ax \ which includes the above, is integrable in finite terms if i being a positive whole number or 0. (J^Ialmsten, Gamlridge Mathematical Journal, 2nd Series, Vol. V. p. 180.) Verify this. 7. As an illustration of the theory of disappearing factors, integrate the equation (^^ + ^^^)5+{(« + 3)2a.^ + (6-z+l)a.}J ■\-[{a + l)q^x-U]u = 0, 8. The equation (1 - ax'^) -i^^ — Ix -^ — cy = 0 is inte- grable in finite terms in the following three cases ; viz. 1st, If - is an odd integer ; 2ndly, If . / jf 1 j + — [ is an odd integer; is an even integer. EXERCISES. 459 9. Integrate tlie partial differential equation dx^ dif X dx 10. The partial differential equation is integrable in finite terms if ^ = . . {Legendre. See Lacroixj Tom. ii. p. 618.) Verify this. 11. Shew that the sum of the series 1.2...?za; + 2.3...(w4-l) ic^..+^ (p + 1) ... {p -\- n - I) x^ may be expressed in the form /dV X^-X^ \dx) 1 — x 12. Sum the series 72 7 13. The equation (a + hx) -^^ + (/+ (jx) -^ + ngu = 0 is integrable in finite terms if n is an integer. Apply the method of Art. 13 to reduce the symbolical equation to a bino- mial form. Or assume a + hx = t. 14. The differential equation can be integrated in finite terms, whatever function of x is represented by Q, (Curtis, Cambridge Mathematical Journal, Vol. IX. p. 280.) 460 EXERCISES. The equation may be expressed in the form \dxj ( x^ ] Let e-'"^'^ u = V', then compare the resvilting form with Ex. 8 of the Chapter. 15. Shew generally that, if we can integrate the equation we can integrate f[-^+ Q]u-}-(J){x)u = X. 16. We meet the equation d'y 1- 3g^ dy 1_ dc' "^ c-c' dc l-c^^-^' in the theory of the elliptic functions (Legendre's modular equation). Shew that it is not integrable in finite terms, but is integrable in the form y = A+B log c, where A and B are series expressed in ascending even powers of c. 17. Prove the following generalization of Prop. iir. 18. Prove the following still more general theorem, ( 461 ) CHAPTER XVIII. SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS BY DEFINITE INTEGRALS. 1. The solution of linear differential equations by definite integrals was first made a direct object of inquiry by Euler. His method consisted in assuming the form of the definite integral, and then, from its properties, determining the class of equations whose solution it is fitted to express. Laplace first devised a method of ascending from the differential equa- tion to the definite integral. And Laplace's is still the most general method of procedure known. Its application is how- ever not wholly free from difficulties, due partly to the present imperfection of the theory of definite integrals, partly to an occasional failure of correspondence in the conditions upon which continuity of form in the differential equation and con- tinuity of form in its solution depend. Indeed it ouglit never to be employed without some means of testing the result a posteriori, e. g. by comparison with the solution of the pro- posed differential equation in series. Frequently indeed it is possible to deduce the solution in definite integrals from the solution in series without employing Laplace's method at all. Laplace's method is applied with peculiar advantage to equations in the coefficients of which x enters only in the first degree, and of which the second member is 0. Expressing any such equation in the form ^*(i)'^+^Glj"='^ ^'^- we must assume w = I ( Tdt, T being a function of t, the form of which, together with the limits of integration, must be determined by substituting the 462 SOLUTION OF LINEAR DIFFERENTIAL [CH. XVIII. expression for u in the proposed differential equation. Effect- ing this substitution, we have a result which may be thus expressed, or, since \xe'''4>{t) Tdt+L''''>lr[t)Tdt = () ...(2). Of this however, the first term is, by integration by parts, reducible to the form Thus, (2) assumes the form e«<^(«)r-/6«{J[<^(0r]-twr}*=o (3), and will therefore be satisfied, if we make ~[{t)T]-^{t)T=0. The former of these equations has reference only to the limits ; the latter, expressed in the form gives on integration, and determines T in the form ^ {t) T= Ce-^"' "JW' ART. 1.] EQUATIONS BY DEFINITE INTEGRALS. 463 Thus, we have u=CJ 6 ^{t) ^' W' the limits of integration "being determined by the equation e ^f^') =0 (5). Should this equation have n distinct roots, these may evidently be so disposed as to give n — 1 distinct particular integrals. Such is the general statement of Laplace's method. Applied to an equation in the coefficients of which the highest power of X involved is the 7i**\ it would make the determination of T depend on the solution of a differential equation of the 'n}^ order. Other practical limitations may be noted. For in- stance, the method is only directly applicable to the expression of integrals which produce on development series of a certain form. Thus, if we develope the exponential in the assumed expression for w, we have u=j Tdt+xj Ttdt + ^2 iTedt + &c. an expansion in which positive and integral powers of x alone present themselves. Integrals of different forms may, however, by preparation of the differential equation, be brought under the dominion of the method. These and other points we pro- pose to illustrate by the detailed examination of a special but very important example, particular forms of which arc of very frequent occurrence in physical inquiries. We shall first, in accordance with what has above been said, determine the different kinds of solution in series of which the equation admits. This part of the investigation is intended to be supplementary to Art. 9 of the last Chapter. -r, r^' d"u du „ Ex. Given x j^ + ^ d^ — 2 ^^^ = 0- 464 SOLUTIONS EXPRESSED BY SERIES. [CH. XVIII. Solutions expressed in Series, 2. The symbolical form of the above equation is "-i>(Z>+.-i)-'^"=Q («)• Hence, if an integral be expressible in the form 'Zu„^x'^, the law of formation of the coefficients u^ will be "' m(w + a-l) ^^^' while the lowest value of m will be 0, or 1 — a. Thus, except in a particular case to be noticed hereafter, the comjDlete in- tegral will be " = -^i^ + 27lTT)+2.4(« + l)(a + 3) + '^'=-' The two series in the general value of u are evidently con- vergent for all values of x. As this question of the conver- gency of series is sometimes important in connexion with the solution of differential equations, the reader is reminded that according as, in the series of terms or groups of terms Uq+U. + U^-^&C, the ratio — ~ tends, when n is indefinitely increased, to a limit less or greater than unity, the series is convergent or divergent; when the ratio is less than unity but tends to unity, we must apply a system of criteria developed by Professor De Morgan (Differential and Integral Calculus, p. 325*). * That this system virtually includes previous special results has been proved by Bertrand {Liouville, Tom. vii. p. 35) ; that it is a legitimate deve- lopment of the fundamental principles of Cauchy has been established by Paucker {Crelle, Band. xlii. p. 138). ART. 2.] SOLUTIONS EXPRESSED BY SERIES. 4C5 When a is an odd integer, the general integral will involve a logaritlim. In particular if a= 1, we shall have u = a^+a,x''-\-a^x'+&c. + logx{h^-\-h^x^+h^x'+&c.) ... (9), a^ and h^ being arbitrary constants, and tlie succeeding coeffi- cients determined by 07i\,, + 2mh,, - c/a^, = 0, m'h^ - q%^,^ =0 (10} . The symbolical equation (6), indicates by its form that there are no solutions expressible in descending powers of x, and infinite in one direction only — i. e. beginning with some finite exponent, and presenting a series of exponents thence descending. But the equation may be transformed so as to admit of a solution of this kind. For, assuming u = e"^\-, we shall liave d^v , ^ . dv and of this the symbolical form will be found to be I){l) + a-l)v-2q{D + ^-l)e'v=0 (11); whence, if v be developed in a scries of the form Sr„j,x"*, the law of derivation of the coefficients will be {m + « - 1) v^ - 2q {m + ^ - 1) v. It follows from this that there will be two ascending and convergent series for v, and one descending and divergent series. The law of the latter series is by changing ni into m + lj more conveniently expressed in the form, (m + 1) (??2 + a) B. D. E. 30 458 SOLUTIONS EXPRESSED BY SERIES. [CH. XYIII. Hence, the first exponent will be — - , and the ultimate value of tt will be If we assume ii = e'^^'v, and proceed as above, we shall obtain for V the symbolical equation, D{D + a-l)v+2q{D + ^--[)6'v = 0 (13), and as this differs from the previous equation for v, only by a change of sign affecting q, we at once deduce a second value of u, in the form u = Ee'''x . jl - -r^-j^ + YTiqV ^"j ^^^^' the terms within the brackets being alternately positive and negative. Both the descending series are finite when a is an even integer, and though for all other values of a they are infinite and ultimately divergent, yet if x be large they begin with being convergent, and may under certain circumstances be employed for numerical calculation. Thus, we have obtained two solutions expressed in ascend- ing series always convergent, and two solutions involving series expressed in descending powers of x, and ultimately divergent. As concerns the convergent series for v, derivable from the transformed equations (11) and (13), we may remark that when multiplied by the developed exponentials, they will only reproduce the convergent series ioru already obtained in (8). One observation yet remains. We have seen that each of the assumptions u = €^% and u=6~^''v, transforms the proposed differential equation into another of which the solution in a ART. 3.] SOLUTION BY DEFINITE INTEGRALS. 467 descending scries is finite wlien the given equation admits of finite integration. Tliis species of transformation is frequently- possible. To accomplish it we must assume u = Qv, the form of Q being determined by the solution of that differential equation upon which, by Props, ii. and iii. Chap, xvii., the solution of the proposed equation, when possible in finite terms, is dependent. Solution of the Equation hy Definite Integrals. 3. Comparing the proposed equation, d^U du o r. / X with the general form (1), we have Hence, Substituting these values in (4), we have u= Cje^'(e-qf''dt (16), while for the limits of integration, (5) gives Hence, supposing a positive, and confining our attention for a tlie present to the factor {t"- q^, which alone determines t in perfect independence of x, we find t= ±q. Thus, u=c{\^'{e-qj"dt, 30—2 468 SOLUTION OF THE EQUATION [CH. XYIII. Assuming then ^ = ^ cos 6, and changing the sign of the arbitrary constant, u=C\ e'''"'^ {sin ey-'cW (17), and this, as its form suggests, and as we shall hereafter shew, is an expression for the particular integral represented by the first convergent series in the general value of u, given in (8). To deduce another integral, let ns in the symbolical equa- tion (6), assume u = e^^~">^v. We find " ^ e'% = 0 (18). {n + i-a)n Hence, a value of v may be determined from tliat of u by changing a — 1 into 1 — a; i.e. by changing a into 2 — a. Thus we have, for the second particular integral, u = C^x'-" ["e^" ^°^ ^ (sin ef" dO, provided that 2 — a he positive. Hence, «/a lie hetween 0 and 2, we have for the complete integral, u= c, [V ^°^^ (sin ey-^do-v c.x'-'' f V^^« (sin ey-'^de . . . (19). If a = l, the two particular integrals in the above expression merge into one. To deduce the true form of the general integral, we may proceed thus, « = ["e^- ^-e {C^ (sin ey-' + C, [x sin (9)'^ dO, - 0 = f > -«{^ (sin er + B (^'"^)°"'-j^^'"^)'"°] de, on replacing C^ and C^ by two new arbitrary constants, A and B, ART. 4.] BY DEFINITE INTEGRALS. 469 Now when a= 1, we find by the usual mode of treating vanishing fractions, (sin(9rM^sin6>)'" 1 r , . c^2^ - — ^_\ — = log [x (sm ey]. Thus, w=["e^-'='^««^[^ + ^log(aj(sin^)"}]fZ(9 (20). This is tlie complete integral of the equation, ^dZ+d:«-2^''='' (2^)' and a similar form exists for all cases in which a is an odd integer. 4. AVe proceed to the cases in which a is fractional and does not lie between the limits 0 and 2. By the application of Props. II. and ill. Chap, xvii., this case can be reduced to the case in which a does lie between the limits 0 and 2. First, suppose a negative ; then we may assume a= a —2n, where a lies between 0 and 2, and n is a positive integer. In this case, the first term of (19) will need transformation. Xow the symbolical equation (6), becomes V(-l)}log^'(sin6')'}^(9 (35). '^ 0 Such is the complete integral. In all physical problems involving partial differential equa- tions the determination of the arbitrary functions so as to satisfy given initial conditions is a matter of great importance, and sometimes, where discontinuity presents itself, of great ART. 9.] paeseval's theohem. 477 difficulty. But tliough some general principles might be stated, the subject is best studied in the concrete application. In applying the above solution to the problem of attraction it is required to determine the arbitrary functions so tliat when r = 0 we should have u = F{z). Now, since, when r = 0, log?- is infinite, it is necessary to suppose -v/r [z) = 0. AVe have then F{z)=rcj>{z)de = 7rcp{z). -' 0 Thus the solution under the proposed limitation becomes u=- rF{z + 7- cos e V(- 1)} de, FarsevaTs Theorem. 9. Equations whose symbolical form is binomial generally admit of solution by definite integrals. PfafF's equation lias thus been treated by Euler. (Lacroix, Tom. in. p. 529.) The very beautiful theorem of Parseval, which makes the limit of the series AA' + BB' + GC + &c. dependent upon the limits B' C of the series A -\- Bu -{- Ciir + &c. and A A f- - .- -1- {€^V(-)} ^ + cj> {e-^V(-^'} ^ (e-^V(-^^}] de (36), vrhicli is the theorem in question. Solution of Differential Equations hy Fourier s Theorem, 10. As Fourier's theorem affords the only general method known for the solution of partial differential equations Avith more than two independent variables (and such are the equa- tions upon which many of the most important problems of mathematical physics depend), we deem it proper to explain at least the principle of this application, referring the reader for a fuller account of it to two memoirs by Cauchy *. As a particular example, let us consider the equation d^u ncifd^u d^u d^viX ^ ,^^, ■W-^\M^-df^di) = '' (^^)- Let u — <^ {x, y, z, t) represent any solution of this equa- tion. By a well-known form of Fourier's theorem, * Sur V Integration d* Equations Lineaires. Exercices d' Analyse et de Phy- sique Mathematiqu£, Tom. I. p. 53. Sur la Transformation et la Reduction des Integrales Generales d'un Systeme d' Equations Lineaires aux differences partielles. Ibid. p. 178. ART. 10.] BY Fourier's theorem. 479 successive applications of wliicli enable us to give to u the form w= i_^ [[III) e^^^"'^^(«, ^, c, i)dailhdcd\diidv (38), — 00 where A = (a - x)\^- i^) — y) y^ ■\- [c — z) v. Substituting this expression in (37), and observing that from the form 2:iven to A we have (S+|+S^^^^-'=^-^^'-"(-^^--^-^')' ^ d' _ d' da we have oo (^ being put for ^ {a, h, c, t). This equation will be satisucd if given in (39) would only lead to an equivalent result. 480 SOLUTION BY FOUETER's THEOREM. [CH. XVIII. To complete the solution, we observe that if, representing 77. + -p + 7;2 ^7 -^j we make t = e^, so as to reduce the given equation to the symbolical form, then, by Propositions 11. and ill. Chap. xvii. the transforma- .. ..dv dv .,, . tionz.= e^^ = ^-, will give D{D-l which is of the same form as the equation for u. Hence, V admitting of expression in the form (40), we have on merely changing the arbitrary function, '' " ^llllli ^'^"^""''^ ^'"'' "^^ ^''' ^' ""^ dadhdcdXdfidv... . (41). — X The complete integral is thus expressed by the sum of the particular integrals (40) and (41). The sextuple integral by which the above particular values of u are expressed admits of reduction to a double integral leading to a form of solution originally obtained by Poisson. Cauchy eifects this reduction by a trigonometrical transformation. It may be accomplished, and perhaps better, by other means; but this is a matter of detail which does not concern the principle of the solution. We may add, that when the function to be integrated becomes infinite within the limits, Cauchy's method of residues should be employed. The reduced integral in its trigonometrical lorm, together witli Poisson's method of solution, which is entirely special, will be found in Gregory's Examples, p. 504. Cauchy's method is directly applicable to equations with second members, and to systems of equations. The above example belongs to the general form d\ T-r ART. 10.] MISCELLANEOUS EXERCISES. 481 where ^ is a function of ->- , -j- , ;t- . For all such equations the method furnishes directly a solution expressed by sextu])le integrals, Avhich are reducible to double integi'als if H is homogeneous and of the second degree. In the above example the double integration proves to be, in eifect, an integration extended over the surface of a sphere whose radius increases uniformly with the time. Integrals of this class are pecu- liarly appropriate for the expression of those physical effects which are propagated through an elastic medium, and leave no trace behind. MISCELLANEOUS EXEECISES. 1. The complete integral of the equation is expressible in the form it, = Ae^"" + Be'^'^, A and B being series which are finite when n is an integer. (Tortolini, Vol. V. p. 161.) 2. The definite integral I cos{7i (d — xsm6)] dd, can be J 0 evaluated when n = ± (t + -j, where ^ is a positive integer or 0. (Liouville, Journal, Tom. Yi. p. 36.) Kepresenting the definite integral by u, it will be found that u satisfies an equation of the fonn -7-5= ( -4 + -2 ) «• The subject of the evaluation of definite integrals by the solution of differ- ential equations has been treated with great generality by Mr Kussell {Philo' sophical Transactions for 1855.) 3. If ^; = a be the equation of a system of curves, v being d^v d% a function of x and y which satisfies the equation -7-3 + -y-:^ = 0, and \i u = ^ be the equation of the orthogonal trajectories of the system, then u may be found by the integration of an B. D. E. 31 482 MISCELLANEOUS EXERCISES. [CH. XVIII. exact differential equation of the first order, and when found will satisfy the equation -^-^ + -^ = 0. The above theorem is applied by Professor Thomson to the problem of determining the forms of the rings and brushes in the spectra produced by biaxal crystals. {Cambridge Journal, 2nd Series, Vol. i. p. 124.) 4. The normal at a point P of a plane curve meets the axis in G, and the locus of the middle point of PG is the parabola y^ = Ix. Find the equation to the curve, supposing it to pass through the origin. ( Cambridge Problems.) 5. The normal at any point of a surface passes through the line represented by y = ^ = - . Find the differential equation to the surface, and obtain the general integral. {lb.) 6. Prove that the differential equation of the surfaces generated by a straight line which passes through the axis of Zy and through a given curve, and which makes a constant angle with the axis of s, is iK£+2/^ = V(^' + 2/Vota. (lb.) 7. Integrate the above equation. 8. Express by a definite integral the series, Form the differential equation by Chap. xvii. Art. 11, and then apply Laplace's method, Chap, xviii. The result is u = — l^cos{x cos 6)d9. (Stokes, CaTiibridge Transactions, Vol. ix. p. 182.) 9. Hence express the series in a form suitable for calcu- lation when X is large. Proceeding according to the directions of Chap, xviir. the complete integral of the differential equation expressed by descending series will be u = x~^{{A cosx + Bsmx)R + {A sin ic - -B cos x) ^}, ]^2 32 ]2 32 52 ^2 where Ji=,i . ___.^+ __^__:_^^_&c. CH. XVIII.] MISCELLANEOUS EXERCISES. 483 The values of A and £ for the particular integral in question will be A =B=Tr~^. These are deduced from the consideration that, when x tends to infinity, we have, in the limit, 2 IT w J ^ cos {x cos 6) do = (ttx)"^ (cos oj + sin x). {Ihid. ) The above series occurs in several physical problems. 10. The complete integral of the equation, ^ li' "^ ^"^ "^ ^^) sf "^ ^^■^•'^^ "^ ^'^'^ ^ "^' may be expressed by a finite formula involving general differ- entiation. (Attributed to Liouville.) Assume y^ze^^ 2~ ; then, by a proper determination of a and /3, the equa- tion may be reduced to the form The symbolical equation obtained by assuming x — e^ will be binomial, and the integration in the required form may be effected by Prop. ill. Chap. xvii. 11. Equations of the form may be reduced to the form, '^S)^+^(3^='^ W' considered in Chap, xvili. Assume x'"=i, y=t^z; the determination of Z; will be foftnd to depend on the equation Jc{J: -1)171^ + k{m{m -1) + mAi} + A(, = 0. Petzval, Lineareii Differentialgldchungen, Pt, 1st, p. 105. Riccati's equa- tion is included in the above. 12. Equations of the form (a,+ h, log x) x' ^, + [a, + \ log x)x-^-\- (a,+ Zi, log a:) w = 0 Are reducible to the form {m), {Ih. p. 112.) 31—2 484 MISCELLANEOUS EXERCISES. [CH. XVIII. 13. The complete integral of the equation •I a where p is a prhnitive root of p""^^ = l, and C, C^, €^...0^, satisfy the condition (7+ (7^+ Cg ... + O„ = 0, but are other- wise arbitrary. (Jacobi, Crelles Journal, Vol, X. p. 279.) 14. The determination of the orthogonal trajectory of any system of straight lines on a plane, involving in their general equation one variable parameter, can be determined by the solution of an exact differential equation between x and y. This interesting proposition, together with the following demonstration, was communicated to the author by Professor Donkin, with whose permission it is published. The equation of the given system can always be expressed in the fonn x sin ^ - y cos ^ = 0 {d)y or, putting cos d = u, sin d = v, vx-uy-F{u^ v) = Q (1), u''-tv^-l = Q (2). The equation of the trajectory will then be udx+vdy=0 (3), u and V being determined from (1) and (2) as functions of x and y. Now, if we represent the first members of (1) and (2) by i^and $ respec- tively, then, in order that (3) may be an exact differential equation, we must have, in virtue of (37) Chap. XIV. dFd^_dF d^ dFd^_dFd^_ dx du du dx dy dv dv dy~~ and this will be found to be identically satisfied. Hence (3) is an exact differ- ential equation, as was to be shewn. The proposition applies generally to the problem of involutes. Thus, the tangents to a circle being represented by TX-uy=a, u^ + v^ = l, the equation (3) "vdll become {x V(a:^ + y^- a^) - ay} dx + {y^/{x^ + y^ - a-} + ax] dy _ x^+y^ This is exact, and determines, on integration, the system of possible involutes. CH. XYIII.] MISCELLANEOUS EXERCISES. 485 15. To determine the connexion of the integrals of any system of simultaneous differential equations expressible in the form dx_dF dji^clF ^ dt die ' dt dv (A du __^dF dv __dF dt dx ' dt dy where i^ is a given function of x. y, u and v. The complete solution will evidently consist of four equations determining ar, y, u, V as functions of t, and four arbitrary constants. Suppose that there exists an integral of the form ^ = c, where •!> is a func- tion of X, y, u, V, not involving t. Then, diflferentiating, we have d^ dx d^ dy d^ du d^ ^^'_n dx dt dy di dv, dt dv dt ' or, substituting for — , -~^ , &,c. the values given in (1), d^dF d^ dF _d^ dF _d^ dF dx du dy dv du dx dv dy Now this equation is identically satisfied if ^ = F. Hence one integral will be F—a, where a is an arbitrary constant. Suppose now that another integral not involving t can be found. Then representing it by $ = 6, and observing that (2) is identical with the equation (4) in the last problem, it is seen that if, from the two equations F=a, = &, vv-e determine u and v as functions of x, y, a, h, the expression ndx + vdy will be an exact differential. Hence, if f{udx + vdy) — x, we have "=% ^=1 <^)- Now differentiating the integral F= a with respect to a, and regarding u, v, as functions of x, y, a, h, we have dF du dF dv du, da dv da * or, putting for , , -^ their values given in (1), and for u, v their values given in (3), d^x^ dx^d^x_ dy^^ dadx dt dady dt ' 486 MISCELLANEOUS EXERCISES. [CH. XYIII. whence, integrating, S=' + ' (*'• c being an arbitrary constant. Since the form of % is known, this constitutes a third integral. Lastly, differentiating F—a with respect to h and proceeding as above, we find db=' ^'^' e being an arbitrary constant. And this is the fourth integral. The above is a simple illustration of the methods of Theoretical Dynamics referred to in Chap. XIV. Thus the equations for the motion of a body attracted towards fixed centres (all in one plane) are d-x_ dR d^y_ dR di^~'~~dx' 'di'~~~dy' R being a function of x, y, and the co-ordinates of the fixed centres. These equations may be expressed in the form dx dy du _ dR dv _ dR dt dx' dt dy' Now, if we represent the function | (u^ + v^) + Rhy F, the above equations assume the general form (1). It was intimated in Chap. XI Y. that the solution of the equations of Dyna- mics is finally dependent on the obtaining of the complete primitive of a non- linear partial difierential equation of the first order ; and this was previously shewn to depend on the integration of an exact differential equation the coeflB- cients of which were determined by the solution of a linear partial differential equation of the first order. Now all this agrees with what has been exemplified above. For the last two integrals, (4) and (.5) are derived, by mere differentia- tion, from x> vv'hile x is found by the integration of an eocact difi'erential equa- tion whose coefficients, u and v, are obtained from equations which satisfy the linear partial difi'erential equation (2). The student is especially referred to the original memoirs by Sir "W. E. Hamilton {On a General Method in Dynamics. Philosophical Transactions, 1834 — 5), to various memoirs by Jacobi contained in his collected works or scattered through Crelle's Journal, and to the recent memoirs of Prof Donkin {On a Class of Differential Equations inchiding those of Dynamics. Philosophi- cal Transactions, 1854 — 5). Liouville's Journal is rich in valuable memoirs on the subject. ( 487 ) ANSWERS. The following table does not contain answers to all the questions proposed in the Exercises, but to a selected number of them, thought amply sutiicient for ordinary requirements. CHAPTER I. 2. (1) 7/=|pa; + V(l+/). (PIere,^. = J). (2) ii-ay=e^\ (3) (1 + a;') ^9 + y = tan"' a:. (4) rrp + 3/ = 2/'loga7. (5) yp"" -\- 2xp == y . (6) y = x2)+{2)). 3. (l)and(2) g + ^,^y = 0. 6. (1) {x-af + (y-bY==l. (2) hx — ay =^ ah {xy — 1) . ^ m '■2m , m,,f 2m\ 8. X Ty — a, y =0, X -z=f[y . jr ' '^. p p" -^ V p J 9. {y — cf — ^.cx, CHAPTER II. 1. (1) \o^xij^rX-y = c. (2) log^-^^=c. (3) (l + a.^)(l+^^ = ca.l W -^(^-^log(l + /)-log{^ + V(l + r)}=^. (5) cos y — c cos X. (6) tan a: tan y — c. 2. Yes. 3. (1) ?/ = ce~^ (2) ?/ = ce"^^^'\ (3) ^'=c^+2c^. (4) a;=ce"""'. (5) Q/+a-)^(7/+2a:)'=c. 488 ANSWERS. 4. (1) x^-xy^-'if+x-y^c, (2) (ij-x-\-\Y[y-\-x-lY=^c. 5. y = (7aj"+r-^-i. 6. (2) y = ax + cx^/{l-x'), (4) 2/ = sina;-l + ce"^'°^ (5) ?/ = taii-'a;-l + ce-^^^'\ 10. il) z=[c^(l-x^)-a]-\ (2) .^ = 06--^--^,. (3) 2 = {ce^^' + |(2a;^+l)}-^ (5) 7/= (c:i? + loga?+ir. CHAPTER III. 1. x'^-^xY+y'=C, 2. x^-y' = cx, 3. x^-rf = cy\ o? + ?/" ?/ - 4. ^ + tan"-=c. 5. a3 + ve^ = c. 6. e"" {x- + ?/^) = c. 7. sin (?2X + my) + cos (??ia7 + ny) = c. 9. V(l+rr^+/) + tan-^- = c, sm-W{x'+y')+sm-'-+e'=c. 10. Assuming? 4 = ^j '^e liave — -, = — \- C ^ X } G — bV a CHAPTER IV. ^' y'-^Y y'^\ f r 4. xyf{x^^xy-f). Complete primitive is x' -\- xy — y"^ = c, 5. (1) Integrating factor, — . Solution, a;^=c^+2c^. X\J \X -\-y j (2) Integrating factor, ^-j^-^^^,. Solution, (2/ + a;)'(y + 2x)' = c. ANSWERS. 489 W ^ = ^/\/(^ + 5)- (^) ^cos| = c. (]. 7/ = ex is the complete primitive. 7- (1) zttt::^- (2) ^ «^,2 ^i/(^i/ + l) xy -^rxy CH2VPTER V. 1. (1) 6^ (2) 1 2. 2/-^ (3) €^andl 4. (2) ,A '■ (3) 2/-V. (4) (l+2/'-a;')- (5) (a;' + y)-^ (C) (x + y + a;^/)-^ (7) {x + y'^) 2 ^2^-2 2\-3 7. If s + P=y tlie equation becomes -j- + 2Pz = — z^, Avliicli is of the general form of 6. 9. When'^g = — .^. Then/(a:)=--^. n dx Q •^ ^ ' (4 CHAPTER VI. Equations 1 to 5 must be reduced to the form (111 x-j- — a]j^ hif = ca;"", of which the solution is according as h and c are like or unlike in sign. In 1 we find 1 = 1, and the solution by (A) is y = a+ — , where y, is given by changing, in the first of the above solutions, a into -a,l into 1, c into 1. In 2, i = 2; apply (A). In 3 apply (D). 490 ANSWERS. _ 7. V(/5'-4a7)+n(z + i) = 0, ^ being any inteo-er posi- tive, negative, or 0. ^' ^^"-(2^^ + l)3/ + ¥ = c^"^', where A is a root of the equation hA^ + A — h = 0. 10. Compare with p. 95. CHAPTER YII. 1. (y - 2aj - c) {7/-dx-c) = 0. 2. (?/-aloga;-c) (y + alogo^-c) =0. 5. Eliminate j:> by means of a log 2^ + 2^/^; + c = x. 12. Complete Primitive 7/ = cx-\- c — c'. Singular Solution ?/ = ^ 4 13. Complete Primitive, y = cx^-\/{lr- aV) . 2 2 Singular Solution, -% + ^=l. 14. cc" + ?/' = co:. 16. Eliminate « by aj = ^-^^ — ^ (c + a sin"^ »). 17. By a;= — -^— y, (c +- + atan~'»). 19. {x-ay+{7j-f{a)Y^l. 21. a^-^/(a)=q/(a)(:r^-l). ANSWERS. 491 CHAPTER VIII. 4. Singular Solution x = a, 6. Differential equation, w = — - — -,. 11. Particular Integral. 13. Singular solution y = 0 ; complete primitive y = 6*"'*'"^^ -x" 16. (1) Envelope species, y = — — . (2) Envelope species, y"^ == 4x^. (3) Not of envelope species, y = x". 17. Singular solution, sJx-\- sjy =^ a, 18. Singular solution, x'^-^y^ — cCK 19. x = cos~^y^~ + {y - y"")^. CHAPTER IX. 19r 4- 7 1. y = ci^+ce'\ 2. y = ci' + ce'''-\- '^^J . 4. ?/ = (Cj + c^ic) cos a? + (("3 + c^a:) sin x. 5. ?/ = ce"'' + (q + c^o?) e^"^. 6. ^ = <"i cos oj + c^ sin a; + (^3 + c^x) e*" + 1. 9. ?/ = ca;'' + - . 10. y = c{x-]- af + c' (x + af, X 492 ANSWERS. 11. y = e'^ {ccos (a^ \/^) +c'sm (ic V^)]. 12. y = ce''^'°-'^+c,6"'^'^-'^ 14. Add x^ to tlie previous value of y. CHAPTER X. 1. ?/ = — — sin a: + c + ex. \/{c+^/y) 4. y = c log x + c. 5. 3/ = CcC" + 9. a; = -log{c?/+/(c)}+c'. 14. 7/ = e--^^^^(//^'^^(7^^+0'). 19. y = c:c. 20. 2/ = -a + i(a6"+ae "). 22. cc + c + (c,^ -/')^ = 0. 23. ?/ = c log {a; + c + V(^' + 2ca?)} + c'. 32. (y_o)^-|; = o. 33. 2/ = lg + a^a.+J ANSWEKS. 493 CHAPTER XL 1. x = cy\ 2. x^c = \\oz{ny^^{inf-\)\. 4. 2cx + c =1 h—a l+a h — a b + aj ' 6. Let if = 2cx — x^ represent the circles, then the tra- jectoiy is x^ = 2c'7/ — y^, 7. y'+ x'^—c = 2a^ log x. 8. An equiangular spiral. 10. 4.a?/ + c = 2ax V(4aV - 1) - log [2ax + VC'iaV - 1)J. CHAPTER XII. 1. {x-a){y-h){z-c) = a 2. x^ + 2f - 6x7/ - 2xz + z^=C. 3. yz ^^ zx ■\- xtj = c. 5. e^iy-]-z)=c. 6. ^ + ^ + ^=(7. X y z X y \ J I 9. a?' + cr?/^ — 2iJ + cc^^ = c. 10. No. CHAPTER XIII. 1. aj = ce ^ - 1 , 7/ = (c^ + cj e ' . 2. ?/ = €~" (c cos ^ + c sin ^ , ic = — - [(c + c') sin i + (c - c) cos ^]. 494 ANSWERS. 5. x + y = ce~" + - + - , O D 7. aj = i + 4c,e^' + 4c,e-'' - Zc/-''^' - c/ iV7 2/ = TT + ^i^ +^^^ - 14 C„6 — C46 CHAPTEE XIV. it 5. z = - + (f){a7/ — hx), 6. z = €''(f){x — y), a y 7. 2 = (x+2/){x+ai/) +.y V(- 1 - a'). CHAPTER XVI. 111 — o??i + 6 3«? sln??2x— (??i^— 2) cos?7ia? ^ o- 10. 2* = COS {n log cc) - THE END. CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESS. [IVERSITY OF CALIFORNIA LIBRAR ENGINEERING AND MATHEMATICAL SCIENCES LIBRARY (213)825-4951 University of California, Los Angeles Please return to the above library NOT LATER THAN DUE DATE stamped below. 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