A TREATISE ON DIFFERENTIAL EQUATIONS

Transcription

1 A TREATISE ON DIFFERENTIAL EQUATIONS A. R. Forsyth Sixth Edition Dover Publications, Inc. Mineola, New York

2 CONTENTS CHAPTER I. INTBODUCTION. ABT Formation of Differential Equations and character of solutions What is to be considered a solution 5 6. Definitions Number of first integrals of a given equation 8 9. Lemma relating to functional dependence Lemma relating to independent equations CHAPTER II. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 11. General equation of the first order An equation of first order and first degree has only one primitive Standard L: variables separable Standard II.: linear equation 20 16, 17. Standard III.: homogeneous equation Standard IV.: one variable absent Standard V.: equation of the n th degree Standard VI.: Clairaut's form 31 22, 23. Existence of Singular Solutions Derivation of the Singular Solution from the primitive , 26. Envelope locus, nodal locus, cuspidal locus Derivation of the Singular Solution from the differential equation; introduction of tao-loous; envelope locus is the only one whose equation is a solution Analytical conditions for a singular solution An equation of the n th degree has not neoessarily a Singular Solution Prinoiple of duality 47 Miscellaneous Examples 50 SCPPLEMENTAEY NOTE : Kunge's method for the numerical solution of differential equations

3 xii CONTENTS CHAPTER III. GENERAL LINEAR EQUATION WITH CONSTANT COEFFICIENTS. AST Theorems in differentiation and integration Form of the linear equation Its primitive consists of two parts General properties Derivation of the Complementary Function Derivation of the Particular Integral in some typical forms , 48. Solution of the homogeneous linear equation Miscellaneous Examples 85 CHAPTER IV. MISCELLANEOUS METHODS. 49. Limitations of methods in the chapter Solution of y( n ) = function of x Solution of y" = function of y Solution of j/(")=function of j/c-d Solution of 2/C>)=function of j/c- 2 ) Depression of order when one variable is absent Equations possessing generalised homogeneity Exact equations which are linear Exact equations which are not linear General linear equation of second order is iutegrable when any single integral of a simpler form is known Reduction of equation to normal form in which the only algebraical coefficient is an invariant , 62. Equation of third order satisfied by quotient of two solutions; the Schwarzian derivative Solution of particular cases of the linear equation by change of independent variable Conditions of equivalence of two given equations Method of Variation of Parameters applied to equations of second order Solution in case of particular form of the invariant Integration by resolution of the differential o p e r a t o r Form of equation used by Sir William Thomson (Lord Kelvin) Condition that a number of particular integrals of the general linear equation should be independent is the non-evanescence of a certain determinant, and value of this determinant Derivation of the Particular Integral by Variation of Parameters Depression of the order when particular integrals are known Solution when all particular integrals but one are known Geometrical application; trajectories 133 Miscellaneous Examples 140

4 CONTENTS xill CHAPTER V. INTEGRATION IN SERIES. 83, 84. Possibility of solution by approximation in the form of a converging series Solution of \<p (x J-)+-^( X J") 2/ = Form of solution when a zero factor enters into the denominator of a coefficient in the series Case in which there is a solution consisting of a finite number of terms LEGENDRE'S equation The solution y = P n The solution y = Q n Different cases to be considered Primitive in the cases when there is only a single particular solution obtained, that is when 2n is an odd integer Differential relation between P n and Q n Modi6ed form of this relation I BBSSEL'S equation ,102. The solutions y = J n and y=j- n I Properties of the functions J The solution y = Y 0 when n is zero The solution y = Y n when n is an iuteger Differential relation between J n and J^n Deduction of Bessel's equation from Legendre's equation BICCATI'S equation. Cases in which this equation and a more general form are integrable in finite terms Effect of knowledge of particular solutions upon the number of quadratures necessary to solve the equation Eeduction of Biccati's equation to Bessel's equation Symbolical solutions 197 Miscellaneous Examples 200 CHAPTER VI. THE HYPERGEOMETRIC SERIES Definition of the series; special cases ,115. Differential equation of the second order satisfied by the series; primitive of the equation Normal form of the differential equation , 118. Equations subsidiary to the deduction of particular solutions of the equation Six values of the variable element , 121. Set of 24 particular s o l u t i o n s

5 XIV CONTENTS 122,123. Division of the 24 solutions into six classes of four each Ranges of significance of the integrals; and linear relations among these integrals Gauss's II function Determination of the constants in the linear relations of The Schwarzian derivative for the differential equation; to be applied to obtain the cases of integration in a finite form Case I.; a;(s» + l) 2 =4s» Oasell.; a;(«4-2s 2 \/3-l)8=(s«+ 28W3-l) Case III.; combination of I. and II References to original memoirs Miscellaneous Examples 240 SUPPLEMENTARY NOTES. I. Integration of linear equations in series by the method of Frobenius, with examples of application to Bessel's equations and others II. Equations having all their integrals regular III. Equations having some (but not all) integrals regular. 265 IV. Equations having no regular integrals; normal integrals, subnormal integrals 269 CHAPTER VII. SOLUTION BY DEFINITE INTEGRALS Applicable to linear equations ,137. Form of integral suitable for x<p (D) y + <// (D) y= General method of determination of the limits Particular form of this method usually applied Proposition relating to the solution, by definite integrals, of the general linear equation Speoial oases of this proposition Form of solution suitable for j/"=xx"y Application to the differential equation of the hypergeometric series Primitive of this equation in the definite-integral form References to memoirs 294 Miscellaneous Examples 294 CHAPTER VIII. ORDINARY EQUATIONS WITH MORE THAN TWO VARIABLES EULER'B equation; Bichelot's method of integration Cauchy's method of integration ,149. Generalisation of Euler's equation; method of integration due to Jacobi 303

6 CONTENTS XV ABT TOTAL DIFFERENTIAL EQUATIONS; formation from given primitive Relation between coefficients in Pdx + Qdy + Edz = 0 that a single primitive should exist; this condition of integrability is sufficient Method of integration when this relation is satisfied Method of integration when this relation is not satisfied Comparison of the primitives in the two cases Geometrical interpretation; the locus represented is a family of curves and sometimes a surface Total equations in n variables; conditions to be satisfied that such an equation should be derivable from a single primitive, and method of integration when conditions are satisfied Method of integration when these conditions are not satisfied; Pfaff's problem Integral equivalent of a total differential equation in three variables, when the condition of integrability ( 152) is not satisfied Modified method for the process in Reduction, to a canonical form, of a total differential equation in four variables, when no condition of integrability is satisfied Integral equivalent of the total equation in 165, Case of equations which are not linear SIMULTANEOUS EQUATIONS; oases in which they arise Method of integration of linear equations with constant coefficients Relations between the arbitrary constants Number of independent arbitrary constants in the general case Forms of solution (i) for imaginary roots, (ii) for equal roots Special forms of solution Simultaneous equations with variable coefficients; sufficient to consider equations of first order When in the modified form there are m dependent variables, the solution can be made to depend upon that of an ordinary equation of the m th order Jacobi's Multipliers Integration of the equations of motion of a particle moving under a central force Miscellaneous Examples 370 CHAPTER IX. PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER Notation and definitions Classification of integrals of a partial differential equation The Complete Integral The Singular Integral The General Integral 878

7 XVI CONTENTS 184. Every solution of the equation (other than a special integral) is included ia some one of the three general classes Geometrical interpretation in the case in which there are two independent variables Derivation of the Singular Integral (if it exists) from the differential equation, with test of existence LAGRANGE'S LINEAR EQUATION ; the differential equation equivalent to0(u,«)=o Derivation of integral of Pp+Qq=S ,190. This integral provides all the integrals that are not special Particular solutions of the equation The form of equations which have an integral <f> {u,v) = Generalisation to the case of n independent variables Homogeneous linear equation in n variables C STANDARD FORMS ,197. Standard I.: \j/ (p, q) 0, and geometrical interpretation ,199. Standard II.: x(2,p, g)=0, and geometrical interpretation Standard III.: <p(x, p) = f(y, g) StandardlV.: z=px + qy + <p(p, q) Duality of partial differential equations This duality corresponds to the principle of duality in geometry Determination, in special cases, of the arbitrary function which occurs in the General Integral Principle of CHAEPIT'S METHOD for the integration of the general equation containing two independent variables Deduction (in two ways) of the subsidiary equations used in Charpit's method Ee-enunciation of the result of The Standard forms are particular cases in which Charpit's method is immediately effective Lagrange's linear equation is a particular case Proof that Standards I., II. and III. are particular cases The GENERAL EQUATION of the first order with n independent variables (Jacobi's method) It can always be changed into an equation which does not contain the dependent variable Principle of the method used by JAOOBI for the integration of the general equation Deduction of the necessary subsidiary equations These subsidiary equations are sufficient Formulation of the rule to which the method leads Lemma on functions connected with the subsidiary equations Integration of the subsidiary equations List of authorities on partial differential equations Examples of Jacobi's method SIMULTANEOUS PARTIAL EQUATIONS Case in which the number of equations given is equal to the number of independent variables 453

8 CONTENTS XVU 231. Case in which the number of equations given is less than the number of independent variables Homogeneous linear systems A complete system remains complete under linear algebraical combinations A method of integration of a complete Jacobian linear system, with examples 461 Miscellaneous Examples 466 CHAPTER X. PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND AND HIGHER ORDERS. 236 Notation and definitions Simple cases of the equation Rr + Ss + Tt=V MONQE'S METHOD of integration of Rr + Ss + Tt=V Investigation of the form of equation to which this process may be applied Deduction of intermediate integral of Rr + Ss + Tt+U (rt -s i )=V When U is zero, two intermediate integrals may be obtained When V is not zero, two intermediate integrals may be obtained Deduction of general integral from any intermediate integral When two intermediate integrals are obtained, they may be treated as simultaneous equations in p and q Proof of the proposition of Summary of the method of solution , 250. Processes to be adopted in failing cases; general examples General method for the construction of an intermediate integral of any form, if such an integral exists Examples of the method Principle of duality LAPLACE'S transformation of the linear equation; one form Two integrable cases of the transformed equation Further transformation when the conditions of 255 are not satisfied Exceptional case POISSON'S method for a special form of the homogeneous equation LINEAR EQUATION WITH CONSTANT COEFFICIENTS The complementary function in the case in which differential coefficients only of the n' h order occur Particular integral in the case of Method of proceeding for the complementary function of the most general form of equation Modification of the. complementary function in special cases., 517

9 xvill CONTENTS 264. Deduction of the particular integral Class of homogeneous equations Miscellaneous methods Solution of the equation =- =a? ^ in two forms at ox i 268. Proof that these two forms are equivalent Solution in the form of a definite integral Solution also by a symbolical method SOLUTION IN SERIES; the Laplace equation = 3 + = ; + =-== ax* 3y 2 3« Special forms of solutions of the Laplace equation AMPERE'S METHOD for equations of the second order Ampere's method; the subsidiary equations; and the combination of integrals , 280. Imschenetsky's generalisation of an integral containing a number of arbitrary constants by the method of variation of parameters 542 Miscellaneous Examples 549 GENERAL EXAMPLES 558 INDEX 581