# A treatise on differential equations

MacMillan, 1954 - 583 pages

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### Contents

 INTRODUCTION 1 Solution of + i 4 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 15 equation of the nth degree 27 Derivation of the Singular Solution from the primitive 35 Principle of duality 45 ABT PAGE 55 Its primitive consists of two parts 61
 Principle of Charpits Method for the integration of the general equation containing two independent variables 376 Deduction of the subsidiary equations used in this method 377 Ecenunciation of the result of 201 378 The Standard forms are particular cases in which Charpits method is immediately effective 380 Lagranges linear equation is a particular case 381 The General Equation of the first order with n independent variables 384 Principle of the method used by Jacobi for the integration of the general equation 385 Deduction of the necessary subsidiary equations 386

 Derivation of the Particular Integral in some typical forms 69 Miscellaneous Examples 81 Form of solution when a zero factor enters into the denominator 86 Solution of yW function of 2_2l 87 Exact equations which are linear 94 Solution of particular cases of the linear equation by change 105 Solution in case of particular form of the invariant 115 Derivation of the Particular Integral by Variation of Parameters 124 Miscellaneous Examples 137 of terms 144 ART PAGE 151 Legendbes equation 155 Differential relation between P and Qn 167 Properties of the functions J 176 AET PAGE 181 Every solution of the equation is included in some one of the three general classes 342 181 Deduction of Bessels equation from Legendres equation 182 Beduction of Biccatis equation to Bessels equation 191 CHAPTER VI 203 ART PAGE 209 Expression for the series with the variable argument made unity 215 The Schwarzian derivative for the differential equation to 222 Case III combination of I and II 230 CHAPTER VII 250 Proposition relating to the solution by definite integrals 256 Application to the differential equation of the hypergeometric 263 CHAPTER VIII 272 Method of integration when this relation is satisfied 289 Identification of this case with that of 153 297 CHAPTER IX 337 independent variables 345 Derivation of the Singular Integral if it exists from the differential equation with tests of existence 347 Lagranges Linear Equation the differential equation equivalent to pu v 0 352 Derivation of integral of Pp + Qq R 354 This integral provides all the integrals that are not singular 355 Particular solutions of the equation 360 Standard Forms 364 jp g 0 and geometrical interpretation 365 xz P ? 0 aQd geometrical interpretation 367 0 x p f y q 369 z px + qy + p q 371 This duality corresponds to the principle of duality in geometry 374 Determination in special cases of the arbitrary function which occurs in the General Integral 375
 These equations are sufficient 388 Formulation of the rule to which the method leads 390 Lemma on functions connected with the subsidiary equations 391 215222 Integration of the subsidiary equations 393 List of authorities on partial differential equations 401 Examples of Jacobis method 402 Simultaneous Partial Equations 406 Case in which the number of equations given is equal to the number of independent variables 407 Case in which the number of equations given is less than the number of independent variables 408 Miscellaneous Examples 411 CHAPTER X 415 Simple cases of the equation Rr + Ss + Tt V 416 Monges Method of integration of Rr + Ss + TtV 417 Deduction of intermediary integral of Rr + Ss + Tt+ Urt 2 V 420 When U is zero two intermediary integrals are in general obtained 421 When U is not zero two intermediary integrals are also in general obtained 422 Deduction of general integral from any intermediary integral 424 Proof of the proposition of 236 425 Summary of the method of solution 428 Processes to be adopted in failing cases 430 Principle of duality 436 Laplaces transformation of the linear equation one form 438 Two integrable cases of the transformed equation 439 Further transformation when the conditions of 244 are not satisfied 440 Exceptional case 442 Poissons method for a special form of the homogeneous equation 443 Linear Equation with Constant Coefficients 444 The complementary function in the case in which differential coefficients only of the nh order occur 445 Particular integral in this case 446 Method of proceeding for the complementary function of the most general form 449 Modification of the complementary function in special cases 451 Deduction of the particular integral 452 Class of homogeneous equations 453 Miscellaneous methods 455 at ox2 257 Proof that these two forms are equivalent 457 Synthetic solution in the form of a definite integral 458 Solution in this form by a symbolical method 459 the equation + + 0 462 Amperes Method for the equation of 232 467 Equations to be satisfied by a function W 469 Index 509