A treatise on differential equations

Front Cover
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

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