Handbook of Exact Solutions for Ordinary Differential Equations

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HANDBOOK OF EXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION......Page 2
Contents......Page 5
Authors......Page 21
Foreword......Page 22
Notations And Some Remarks......Page 23
0.1.1. General Concepts. The Cauchy Problem. Uniqueness And Existence Theorems......Page 25
0.1.2. Equations Solved For The Derivative. Simplest Techniques Of Integration......Page 27
0.1.3. Exact Differential Equations. Integrating Factor......Page 29
0.1.4. Riccati Equation......Page 30
0.1.5. Abel Equations Of The First Kind......Page 33
0.1.6. Abel Equations Of The Second Kind......Page 34
0.1.7. Equations Not Solved For The Derivative......Page 38
0.1.8. Contact Transformations......Page 39
0.1.9. Approximate Analytic Methods For Solution Of Equations......Page 42
0.1.10. Numerical Integration Of Differential Equations......Page 44
0.2.1. Formulas For The General Solution. Some Transformations......Page 45
0.2.2. Representation Of Solutions As A Series In The Independent Variable......Page 47
0.2.3. Asymptotic Solutions......Page 48
0.2.4. Boundary Value Problems......Page 51
0.2.5. Eigenvalue Problems......Page 53
0.3.1. Form Of The General Solution. Cauchy Problem......Page 57
0.3.2. Equations Admitting Reduction Of Order......Page 58
0.3.3. Methods Of Regular Series Expansions With Respect To The Independent Variable Or Small Parameter......Page 61
0.3.4. Perturbation Methods Of Mechanics And Physics......Page 64
0.3.5. Galerkin Method And Its Modifications (projection Methods)......Page 70
0.3.6. Iteration And Numerical Methods......Page 73
0.4.1. Linear Equations With Constant Coefficients......Page 75
0.4.2. Linear Equations With Variable Coefficients......Page 76
0.4.3. Asymptotic Solutions Of Linear Equations......Page 78
0.5.2. Equations Admitting Reduction Of Order......Page 80
0.5.3. A Method For Construction Of Solvable Equations Of General Form......Page 83
0.6.1. Lie Group Method. Point Transformations......Page 84
0.6.2. Contact Transformations. B¨acklund Transformations. Formal Operators. Factorization Principle......Page 89
0.6.3. First Integrals (conservation Laws)......Page 95
0.6.4. Discrete-group Method. Point Transformations......Page 97
0.6.5. Discrete-group Method. The Method Of Rf-pairs......Page 99
1.1.5. Bernoulli Equation......Page 104
1.2.2. Equations Containing Power Functions......Page 105
1.2.3. Equations Containing Exponential Functions......Page 112
1.2.4. Equations Containing Hyperbolic Functions......Page 115
1.2.5. Equations Containing Logarithmic......Page 117
1.2.6. Equations Containing Trigonometric Functions......Page 118
1.2.7. Equations Containing Inverse Trigonometric Functions......Page 123
1.2.8. Equations With Arbitrary Functions......Page 125
1.2.9. Some Transformations......Page 128
1.3.1. Equations Of The Form......Page 130
1.3.2. Equations Of The Form......Page 143
1.3.3. Equations Of The Form......Page 144
1.3.4. Equations Of The Form......Page 155
1.3.5. Some Types Of First- And Second-order Equations Reducible To Abel Equations Of The Second Kind......Page 159
1.4.1. Abel Equations Of The First Kind......Page 161
1.4.2. Equations Of The From......Page 165
1.4.3. Equations Of The Form......Page 167
1.4.4. Equations Of The Form......Page 174
1.4.5. Equations Of The Form......Page 178
1.5.1. Equations Containing......Page 182
1.5.2. Equations Containing Exponential Functions......Page 185
1.5.3. Equations Containing Hyperbolic Functions......Page 189
1.5.4. Equations Contaning Logarithmic Functions......Page 191
1.5.5. Equations Containing Trigonometric Functions......Page 192
1.5.6. Equations Containing Combinations Of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions......Page 194
1.6.1. Equations Of The Second Degree in y'x......Page 196
1.6.2. Equations Of The Third Degree In y'x......Page 203
1.6.3. Equations Of The Form (y'x)k=f(y)+g(x)......Page 205
1.6.4. Other Equations......Page 213
1.7.1. Equations Containing Power Functions......Page 218
1.7.2. Equations Containing Exponential And Hyperbolic Functions......Page 220
1.7.3. Equations Containing Logarithmic Functions......Page 222
1.7.4. Equations Containing Trigonometric Functions......Page 223
1.7.5. Equations Containing Combinations Of Exponential, Logarithmic, And Trigonometric Functions......Page 224
1.8.1. Some Equations......Page 226
1.8.2. Some Transformations......Page 235
2.1.2. Equations Containing Power Functions......Page 236
2.1.3. Equations Containing Exponential Functions......Page 269
2.1.4. Equations Containing Hyperbolic Functions......Page 275
2.1.5. Equations Containing Logarithmic Functions......Page 280
2.1.6. Equations Containing Trigonometric Functions......Page 283
2.1.7. Equations Containing Inverse Trigonometric Functions......Page 294
2.1.8. Equations Containing Combinations Of Exponential, Logarithmic, Trigonometric,............Page 300
2.1.9. Equations With Arbitrary Functions......Page 308
2.1.10. Some Transformations......Page 315
2.2.1. Equations Of The Form......Page 318
2.2.2. Equations Of The Form......Page 322
2.2.3. Lienard Equations......Page 324
2.2.4. Rayleigh Equations......Page 327
2.3.1. Exact Solutions......Page 329
2.3.2. First Integrals (conservation Laws)......Page 335
2.4.1. Classification Table......Page 337
2.4.2. Exact Solutions......Page 341
2.5.1. Classification Table......Page 359
2.5.2. Exact Solutions......Page 362
2.5.3. Some Formulas And Transformations......Page 377
2.6.1. Modified Emden–fowler Equation......Page 379
2.6.2. Equations Of The Form......Page 388
2.6.3. Equations Of The Form......Page 416
2.6.4. Other Equations......Page 429
2.7. Equations Of The Form......Page 434
2.7.1. Equations Of The Form......Page 435
2.7.2. Equations Containing Power Functions......Page 437
2.7.3. Equations Containing Exponential Functions......Page 441
2.7.4. Equations Containing Hyperbolic Functions......Page 444
2.7.5. Equations Containing......Page 446
2.7.6. Some Transformations......Page 447
2.8.1. Equations Containing Power Functions......Page 448
2.8.2. Painlev´e Transcendents......Page 455
2.8.3. Equations Containing Exponential Functions......Page 461
2.8.4. Equations Containing Hyperbolic Functions......Page 468
2.8.5. Equations Containing Logarithmic Functions......Page 473
2.8.6. Equations Containing Trigonometric Functions......Page 475
2.8.7. Equations Containing The Combinations Of Exponential, Hyperbolic, Logarithmic, And Trigonometric Functions......Page 482
2.9.1. Equations Of The Form......Page 484
2.9.2. Equations Of The Form......Page 490
2.9.3. Equations Of The Form......Page 494
2.9.4. Equations Of The Form......Page 498
2.9.5. Equations Not Solved For Second Derivative......Page 507
2.9.6. Equations Of General Form......Page 509
2.9.7. Some Transformations......Page 515
3.1.1. Preliminary Remarks......Page 518
3.1.2. Equations Containing Power Functions......Page 519
3.1.3. Equations Containing Exponential Functions......Page 535
3.1.4. Equations Containing Hyperbolic Functions......Page 539
3.1.5. Equations Containing Logarithmic Functions......Page 548
3.1.6. Equations Containing Trigonometric Functions......Page 551
3.1.7. Equations......Page 562
3.1.8. Equations Containing Combinations Of Exponential,......Page 567
3.1.9. Equations Containing Arbitrary Functions......Page 573
3.2.1. Classification Table......Page 582
3.2.2. Equations Of The Form......Page 589
3.2.3. Equations Of The Form......Page 590
3.2.4. Equations With......Page 591
3.3.1. Equations Containing Power Functions......Page 615
3.3.2. Equations Containing Exponentialfunctions......Page 618
3.3.3. Other Equations......Page 622
3.4.1. Equations Containing Power Functions......Page 624
3.4.2. 3.4.2.equationscontainingexponentialfunctions......Page 631
3.4.3. 3.4.3.equationscontaininghyperbolicfunctions......Page 634
3.4.4. Equations Containing Logarithmic Functions......Page 639
3.4.5. Equationscontainingtrigonometricfunctions......Page 641
3.5.1. Equations Of The Form......Page 645
3.5.2. Equationsof The Form......Page 647
3.5.3. Equations Of The Form......Page 652
3.5.4. Equations Of The Form......Page 657
3.5.5. Other Equations......Page 659
4.1.2. Equations Containing Power Functions......Page 663
4.1.3. Equations Containing Exponential And Hyperbolic Functions......Page 670
4.1.4. Equations Containing Logarithmic Functions......Page 673
4.1.5. Equations Containing Trigonometric Functions......Page 674
4.1.6. Equations Containing Arbitrary Functions......Page 677
4.2.1. Equations Containing Power Functions......Page 681
4.2.2. Equations Containing Exponential Functions......Page 688
4.2.3. Equations Containing Hyperbolic Functions......Page 690
4.2.4. Equations Containing Logarithmic Function......Page 694
4.2.5. Equations Containing Trigonometric Functions......Page 696
4.2.6. Equations Containing Arbitrary Functions......Page 700
5.1.2. Equations Containing Power Functions......Page 711
5.1.3. Equations Containing Exponential And Hyperbolic Functions......Page 717
5.1.5. Equations Containing Trigonometric Functions......Page 720
5.1.6. Equations Containing Arbitrary Functions......Page 723
5.2.1. Equations Containing Power Functions......Page 727
5.2.2. Equations Containing Exponential Functions......Page 733
5.2.3. Equations Containing Hyperbolic Functions......Page 735
5.2.4. Equations Containing Logarithmic Functions......Page 739
5.2.5. Equations Containing Trigonometric Functions......Page 740
5.2.6. Equations Containing Arbitrary Functions......Page 744
S.1.1. Trigonometric Functions......Page 756
S.1.2. Hyperbolicfunctions......Page 759
S.1.3. Inverse Trigonometric Functions......Page 761
S.1.4. Inverse Hyperbolic Functions......Page 763
S.2.1. Some Symbols And Coefficients......Page 764
S.2.2. Error Functions And Exponential Integral......Page 765
S.2.3. Gamma And Beta Functions......Page 766
S.2.4. Incomplete Gamma And Beta Functions......Page 768
S.2.5. Bessel Functions......Page 769
S.2.6. Modified Bessel Functions......Page 772
S.2.7. Degenerate Hypergeometric Functions......Page 774
S.2.8. Hypergeometric Functions......Page 776
S.2.9. Legendre Functions And Legendre Polynomials......Page 777
S.2.10. Parabolic Cylinder Functions......Page 779
S.2.11. Orthogonal Polynomials......Page 780
S.2.12. The Weierstrass Function......Page 783
S.3.1. Integrals Containing Rational Fonctions......Page 784
S.3.2. Integrals Containing Irrational Functions......Page 788
S.3.4. Integrals Containing Hyperbolic Functions......Page 790
S.3.5. Integrals Containing Logarithmic Functions......Page 793
S.3.6. Integrals Containing Trigonometric Functions......Page 794
S.3.7. Integrals Containing Inverse Trigonometric Functions......Page 798
References......Page 800