Exact solutions of differential equations continue to play an important role in the understanding of many phenomena and processes throughout the natural sciences in that they can verify the correctness of or estimate errors in solutions reached by numerical, asymptotic, and approximate analytical methods. The new edition of this bestselling handbook now contains the exact solutions to more than 6200 ordinary differential equations. The authors have made significant enhancements to this edition, including:В· An introductory chapter that describes exact, asymptotic, and approximate analytical methods for solving ordinary differential equations В· The addition of solutions to more than 1200 nonlinear equationsВ· An improved format that allows for an expanded table of contents that makes locating equations of interest more quickly and easilyВ· Expansion of the supplement on special functionsThis handbook's focus on equations encountered in applications and on equations that appear simple but prove particularly difficult to integrate make it an indispensable addition to the arsenals of mathematicians, scientists, and engineers alike.
Completed with index, and correct pagination.
Author(s): Andrei D. Polyanin, Valentin F. Zaitsev
Edition: 2
Publisher: Chapman & Hall/CRC
Year: 2003
Language: English
Pages: XXVI; 787
City: Boca Raton
HANDBOOK OF EXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION
Cover
Title Page
Contents
Authors
Foreword
Notations And Some Remarks
Some Definitions, Formulas, Methods, And Transformations
0.1. First-order Differential Equations
0.1.1. General Concepts. The Cauchy Problem. Uniqueness And Existence Theorems
0.1.2. Equations Solved For The Derivative. Simplest Techniques Of Integration
0.1.3. Exact Differential Equations. Integrating Factor
0.1.4. Riccati Equation
0.1.5. Abel Equations Of The First Kind
0.1.6. Abel Equations Of The Second Kind
0.1.7. Equations Not Solved For The Derivative
0.1.8. Contact Transformations
0.1.9. Approximate Analytic Methods For Solution Of Equations
0.1.10. Numerical Integration Of Differential Equations
0.2. Second-order Linear Differential Equations
0.2.1. Formulas For The General Solution. Some Transformations
0.2.2. Representation Of Solutions As A Series In The Independent Variable
0.2.3. Asymptotic Solutions
0.2.4. Boundary Value Problems
0.2.5. Eigenvalue Problems
0.3. Second-order Nonlinear Differential Equations
0.3.1. Form Of The General Solution. Cauchy Problem
0.3.2. Equations Admitting Reduction Of Order
0.3.3. Methods Of Regular Series Expansions With Respect To The Independent Variable Or Small Parameter
0.3.4. Perturbation Methods Of Mechanics And Physics
0.3.5. Galerkin Method And Its Modifications (projection Methods)
0.3.6. Iteration And Numerical Methods
0.4. Linear Equations Of Arbitrary Order
0.4.1. Linear Equations With Constant Coefficients
0.4.2. Linear Equations With Variable Coefficients
0.4.3. Asymptotic Solutions Of Linear Equations
0.5. Nonlinear Equations Of Arbitrary Order
0.5.1. Structure Of The General Solution. Cauchy Problem
0.5.2. Equations Admitting Reduction Of Order
0.5.3. A Method For Construction Of Solvable Equations Of General Form
0.6. Lie Group And Discrete-group Methods
0.6.1. Lie Group Method. Point Transformations
0.6.2. Contact Transformations. B¨acklund Transformations. Formal Operators. Factorization Principle
0.6.3. First Integrals (conservation Laws)
0.6.4. Discrete-group Method. Point Transformations
0.6.5. Discrete-group Method. The Method Of Rf-pairs
Chapter 1: First-order Differential Equations
1.1. Simplest Equations With Arbitrary Functions Integrable In Closed Form*
1.1.1. Equations Of The Form
1.1.2. Equations Of The Form
1.1.3. Separable Equations
1.1.4. Linear Equation
1.1.5. Bernoulli Equation
1.1.6. Homogeneous Equation
1.2. Riccati Equation
1.2.1. Preliminary Remarks
1.2.2. Equations Containing Power Functions
1.2.3. Equations Containing Exponential Functions
1.2.4. Equations Containing Hyperbolic Functions
1.2.5. Equations Containing Logarithmic
1.2.6. Equations Containing Trigonometric Functions
1.2.7. Equations Containing Inverse Trigonometric Functions
1.2.8. Equations With Arbitrary Functions
1.2.9. Some Transformations
1.3. Abel Equations Of The Second Kind
1.3.1. Equations Of The Form
1.3.2. Equations Of The Form
1.3.3. Equations Of The Form
1.3.4. Equations Of The Form
1.3.5. Some Types Of First- And Second-order Equations Reducible To Abel Equations Of The Second Kind
1.4. Equations Containing Polynomial Functions Of
1.4.1. Abel Equations Of The First Kind
1.4.2. Equations Of The From
1.4.3. Equations Of The Form
1.4.4. Equations Of The Form
1.4.5. Equations Of The Form
1.5. Equations Of The Form ...... Containing Arbitrary
1.5.1. Equations Containing
1.5.2. Equations Containing Exponential Functions
1.5.3. Equations Containing Hyperbolic Functions
1.5.4. Equations Contaning Logarithmic Functions
1.5.5. Equations Containing Trigonometric Functions
1.5.6. Equations Containing Combinations Of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions
1.6. Equations Of The Form ( , , ) = 0 Containing Arbitrary Parameters
1.6.1. Equations Of The Second Degree in y'x
1.6.2. Equations Of The Third Degree In y'x
1.6.3. Equations Of The Form (y'x)k=f(y)+g(x)
1.6.4. Other Equations
1.7. Equations Of The Form ( , ) = ( , ) Containing Arbitrary Functions
1.7.1. Equations Containing Power Functions
1.7.2. Equations Containing Exponential And Hyperbolic Functions
1.7.3. Equations Containing Logarithmic Functions
1.7.4. Equations Containing Trigonometric Functions
1.7.5. Equations Containing Combinations Of Exponential, Logarithmic, And Trigonometric Functions
1.8. Equations Of The Form ( , , ) = 0 Containing Arbitrary Functions
1.8.1. Some Equations
1.8.2. Some Transformations
Chapter 2: Second-order Differential Equations
2.1. Linear Equations
2.1.1. Representation Of The General Solution Through A Particular Solution
2.1.2. Equations Containing Power Functions
2.1.3. Equations Containing Exponential Functions
2.1.4. Equations Containing Hyperbolic Functions
2.1.5. Equations Containing Logarithmic Functions
2.1.6. Equations Containing Trigonometric Functions
2.1.7. Equations Containing Inverse Trigonometric Functions
2.1.8. Equations Containing Combinations Of Exponential, Logarithmic, Trigonometric,......
2.1.9. Equations With Arbitrary Functions
2.1.10. Some Transformations
2.2. Autonomous Equations
2.2.1. Equations Of The Form
2.2.2. Equations Of The Form
2.2.3. Lienard Equations
2.2.4. Rayleigh Equations
2.3. Emden Fowler Equation
2.3.1. Exact Solutions
2.3.2. First Integrals (conservation Laws)
2.3.3. Some Formulas And Transformations
2.4. Equations Of The Form
2.4.1. Classification Table
2.4.2. Exact Solutions
2.5. Generalized Emden Fowler Equation
2.5.1. Classification Table
2.5.2. Exact Solutions
2.5.3. Some Formulas And Transformations
2.6. Equations Of The Form
2.6.1. Modified Emden–fowler Equation
2.6.2. Equations Of The Form
2.6.3. Equations Of The Form
2.6.4. Other Equations
2.7. Equations Of The Form
2.7.1. Equations Of The Form
2.7.2. Equations Containing Power Functions
2.7.3. Equations Containing Exponential Functions
2.7.4. Equations Containing Hyperbolic Functions
2.7.5. Equations Containing
2.7.6. Some Transformations
2.8. Some Nonlinear Equations With Arbitrary Parameters
2.8.1. Equations Containing Power Functions
2.8.2. Painlev´e Transcendents
2.8.3. Equations Containing Exponential Functions
2.8.4. Equations Containing Hyperbolic Functions
2.8.5. Equations Containing Logarithmic Functions
2.8.6. Equations Containing Trigonometric Functions
2.8.7. Equations Containing The Combinations Of Exponential, Hyperbolic, Logarithmic, And Trigonometric Functions
2.9. Equations Containing Arbitrary Functions
2.9.1. Equations Of The Form
2.9.2. Equations Of The Form
2.9.3. Equations Of The Form
2.9.4. Equations Of The Form
2.9.5. Equations Not Solved For Second Derivative
2.9.6. Equations Of General Form
2.9.7. Some Transformations
Chapter 3: Third-order Differential Equations
3.1. Linear Equations
3.1.1. Preliminary Remarks
3.1.2. Equations Containing Power Functions
3.1.3. Equations Containing Exponential Functions
3.1.4. Equations Containing Hyperbolic Functions
3.1.5. Equations Containing Logarithmic Functions
3.1.6. Equations
3.1.6. Equations Containing Trigonometric Functions
3.1.7. Equations
3.1.8. Equations Containing Combinations Of Exponential,
3.1.9. Equations Containing Arbitrary Functions
3.2. Equations Of The Form
3.2.1. Classification Table
3.2.2. Equations Of The Form
3.2.3. Equations Of The Form
3.2.4. Equations With
3.2.5. Some Transformations
3.3. Equations Of The Form
3.3.1. Equations Containing Power Functions
3.3.2. Equations Containing Exponentialfunctions
3.3.3. Other Equations
3.4. Nonlinear Equations With Arbitrary Parameters
3.4.1. Equations Containing Power Functions
3.4.2. 3.4.2.equationscontainingexponentialfunctions
3.4.3. 3.4.3.equationscontaininghyperbolicfunctions
3.4.4. Equations Containing Logarithmic Functions
3.4.5. Equationscontainingtrigonometricfunctions
3.5. Nonlinear Equations Containing Arbitrary Functions
3.5.1. Equations Of The Form
3.5.2. Equationsof The Form
3.5.3. Equations Of The Form
3.5.4. Equations Of The Form
3.5.5. Other Equations
Chapter 4: Fourth-order Differential Equations
4.1. Linear Equations
4.1.1. Preliminary Remarks
4.1.2. Equations Containing Power Functions
4.1.3. Equations Containing Exponential And Hyperbolic Functions
4.1.4. Equations Containing Logarithmic Functions
4.1.5. Equations Containing Trigonometric Functions
4.1.6. Equations Containing Arbitrary Functions
4.2. Nonlinear Equations
4.2.1. Equations Containing Power Functions
4.2.2. Equations Containing Exponential Functions
4.2.3. Equations Containing Hyperbolic Functions
4.2.4. Equations Containing Logarithmic Function
4.2.5. Equations Containing Trigonometric Functions
4.2.5. Equations Containing Trigonometric Functions
4.2.6. Equations Containing Arbitrary Functions
Chapter 5: Higher-order Differential Equations
5.1. Linear Equations
5.1.1. Preliminary Remarks
5.1.2. Equations Containing Power Functions
5.1.3. Equations Containing Exponential And Hyperbolic Functions
5.1.4. Equations Containing Logarithmic Functions
5.1.6. Equations Containing Arbitrary Functions
5.2.1. Equations Containing Power Functions
5.1.5. Equations Containing Trigonometric Functions
5.1.6. Equations Containing Arbitrary Functions
5.2. Nonlinear Equations
5.2.1. Equations Containing Power Functions
5.2.2. Equations Containing Exponential Functions
5.2.3. Equations Containing Hyperbolic Functions
5.2.4. Equations Containing Logarithmic Functions
5.2.5. Equations Containing Trigonometric Functions
5.2.6. Equations Containing Arbitrary Functions
Supplements
S.1. Elementary Functions And Their Properties
S.1.1. Trigonometric Functions
S.1.2. Hyperbolicfunctions
S.1.3. Inverse Trigonometric Functions
S.1.4. Inverse Hyperbolic Functions
S.2. Special Functions And Their Properties
S.2.1. Some Symbols And Coefficients
S.2.2. Error Functions And Exponential Integral
S.2.3. Gamma And Beta Functions
S.2.4. Incomplete Gamma And Beta Functions
S.2.5. Bessel Functions
S.2.6. Modified Bessel Functions
S.2.7. Degenerate Hypergeometric Functions
S.2.8. Hypergeometric Functions
S.2.9. Legendre Functions And Legendre Polynomials
S.2.10. Parabolic Cylinder Functions
S.2.11. Orthogonal Polynomials
S.2.12. The Weierstrass Function
S.3. Tables Of Indefinite Integrals
S.3.1. Integrals Containing Rational Fonctions
S.3.2. Integrals Containing Irrational Functions
S.3.3. Integrals Containing Exponential Functions
S.3.4. Integrals Containing Hyperbolic Functions
S.3.5. Integrals Containing Logarithmic Functions
S.3.6. Integrals Containing Trigonometric Functions
S.3.7. Integrals Containing Inverse Trigonometric Functions
References
Index
