## A treatise on differential equations |

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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 |