## A Treatise on Differential EquationsSixth edition (1928) of classic 19th-century work considered one of the finest treatments of the topic. Differential equations of the 1st order, general linear equations with constant coefficients, integration in series, hypergeometric series, solution by definite integrals, many other topics. Over 800 examples. Index. |

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### Contents

CHAPTER I | 1 |

Lemma relating to functional dependence | 11 |

CHAPTER II | 17 |

one variable absent | 27 |

Derivation of the Singular Solution from the primitive | 37 |

An equation of the ntb degree haa not necessarily a Singular | 43 |

Its primitive consists of two parts | 63 |

Derivation of the Particular Integral in some typical forma | 71 |

Total equations in n variables conditions to be satisfied that such | 324 |

Modified method for the process in 152 | 331 |

SIMULTANEOUS EQUATIONS cases in which they arise | 342 |

Simultaneous equations with variable coefficients sufficient to con | 350 |

Jacobis Multipliers | 356 |

Integration of the equations of motion of a particle moving under | 370 |

included in some one of the three general classes | 379 |

18L The Complete Integral | 382 |

Solationof | 85 |

CHAPTER IV | 87 |

Equations possessing generalised homogeneity | 94 |

General linear equation of second order i iutegrable when any single | 101 |

Solution of particular cases of the linear equation by change of inde | 108 |

Solution in case of particular form of the invariant | 118 |

122123 Division of the 24 aohuions into six classes of four each 216 | 124 |

Derivation of the Particular Integral by Variation of Parometera | 127 |

AST PAOE | 151 |

LEGEXDREs equation | 159 |

Different cases to be considered | 165 |

Differential relation oetween Pnand Qn | 171 |

Properties of the functions J | 180 |

Deduction of Bessels equation from Legendres equation | 186 |

Reduction of Riccatia equation to Bessels equation | 195 |

CHAPTER VI | 207 |

Six values of the variable element | 213 |

these integrals | 218 |

Gausss 11 function | 221 |

126129 Determination of the constants in the linear relations of 124 | 224 |

The Schwarzian derivative for the differential equation to be applied to obtain the cases of integration in a finite form | 230 |

ART FAOE | 231 |

Case I ic + la4 | 232 |

Case II i c 2s s3 ia + 2 Jf I | 233 |

References to original memoirs | 239 |

Miscellaneous Examples | 240 |

SUPPLXMSNTARY NOTZS I Integration of linear equations in series by the method of Frobenius with examples of application to BessePs equations... | 243 |

Equations having all their integrals regular | 258 |

Equations having some but not all integrals regular | 265 |

Equations having no regular integrals normal integrals subnormal integrals | 275 |

SOLUTION BY DEFINITE INTEGRALS 135 Applicable to linear equations | 277 |

General method of determination of the limits | 279 |

Particular form of this method usually applied | 280 |

Proposition relating to the solution by definite integrals of the general linear equation | 283 |

Special cases of this proposition | 286 |

Form of solution suitable for y xy | 288 |

Application to the differential equation of the hypergeometric series | 290 |

Primitive of this equation in the definiteintegral form | 292 |

References to memoirs | 294 |

ORDINARY EQUATIONS WITH MORE THAN TWO VARIABLES 146 ETLEKS equation Richelots method of integration | 299 |

Method of integration when this relation is satisfied | 314 |

Geometrical interpretation in the case in which there are two independent variables | 384 |

Derivation of the Singular Integral if it exists from the differen tial equation with test of existence | 387 |

Derivation of integral of Pp + QqR | 393 |

189190 This integral provides all the integrals that are not special | 394 |

Particular solutions of the equation | 401 |

Homogeneous linear equation in n variables | 407 |

xzp ? 0 and geometrical interpretation | 410 |

iIx p y q | 412 |

zpx + qy + ipp q | 414 |

Duality of partial differential equations | 415 |

This duality corresponds to the principle of duality in geometry | 417 |

Determination in special cases of the arbitrary function which occurs in the General Integral | 418 |

Principle of CHAHPITS METHOD for the integration of the general equation containing two independent variables | 420 |

Eeenunciation of the result of 206 | 421 |

The Standard forms are particular cases in which Charpits method is immediately effective | 426 |

Lagranges linear equation is a particular case | 427 |

211213 Proof that Standards I II and III are particular cases | 430 |

Principle of the method naed by JACOSI for the integration of the general equation | 431 |

Deduction of the necessary subsidiary equations | 432 |

These subsidiary equations are sufficient | 434 |

Formulation of the rule to which the method leads | 436 |

Lemma on functions connected with the subsidiary equations | 437 |

221227 Integration of the subsidiary equations | 439 |

List of authorities on partial differential equations | 447 |

Examples of Jacobis method | 448 |

SIMULTANROUS PARTIAL EQUATIONS | 453 |

CHAPTER X | 470 |

Cauchys method of integration 301 | 473 |

Deduction of intermediate integral of Rr + Ss+ Tt+ Urtt2V | 476 |

Summary of the method of solution | 484 |

General method for the construction of an intermediate integral | 493 |

Principle of duality | 503 |

POISBOSS method for a special form of the homogeneous equation 609 | 511 |

Modification of the complementary function in special cases | 517 |

Class of homogeneous equations | 520 |

Solution also by a symbolical method | 526 |

AuptREs METHOD for equations of the second order | 533 |

GENERAL EXAMPLES | 558 |

581 | |

582 | |

### Common terms and phrases

arbitrary constants arbitrary function ascending powers assume complementary function complete integral consider contains converging corresponding deduced definite integrals denoted dependent derived differential coefficients dx dx dx dy elimination equal equation becomes equation giving equation satisfied equivalent expansions in ascending expressed in terms finite form foregoing Frobenius method given equation Hence hypergeometric series independent integrals independent variable indicated indicial equation infinite integral equation Integrate the equation intermediate integral latter left-hand side Legendre's equation linear equation memoir method multiplied normal integral Obtain the primitive occur ordinary differential equations original equation orthogonal trajectory partial differential equation particular integral particular solution positive integer preceding proceed Prove quantities regular integral relation right-hand side roots satisfy the equation second order Shew simultaneous equations Singular Integral singular solution Solve the equations subsidiary equations substituted suppose surface tangent term involving theorem transformed unity vanish write