Handbook of Differential Equations

Through the previous three editions, Handbook of Differential Equations has proven an invaluable reference for anyone working within the field of mathematics, including academics, students, scientists, and professional engineers.

The book is a compilation of methods for solving and approximating differential equations. These include the most widely applicable methods for solving and approximating differential equations,　as well as　numerous methods. Topics include methods for ordinary differential equations, partial differential equations, stochastic differential equations, and systems of such equations.

Included for nearly every method are:

• The types of equations to which the method is applicable
• The idea behind the method
• The procedure for carrying out the method
• At least one simple example of the method
• Any cautions that should be exercised
• Notes for more advanced users

The fourth edition includes corrections, many supplied by readers, as well as many new methods and techniques. These new and corrected entries make necessary improvements in this edition.

Table of Contents

I.A Definitions and Concepts.

1. Definition of Terms. 2. Alternative Theorems. 3. Bifurcation Theory. 4. Chaos in Dynamical Systems. 5. Classification of Partial Differential Equations. 6. Compatible Systems. 7. Conservation Laws. 8. Differential Equations - Diagrams. 9. Differential Equations - Symbols. 10. Differential Resultants. 11. Existence and Uniqueness Theorems. 12. Fixed Point Existence Theorems. 13. Hamilton - Jacobi Theory. 14. Infinite Order Differential Equations. 15. Integrability of Systems. 16. Inverse Problems. 17. Limit Cycles. 18. PDEs & Natural Boundary Conditions. 19. Normal Forms: Near-Identity Transformations. 20. q-Differential Equations. 21. Quaternionic Differential Equations. 22. Self-Adjoint Eigenfunction Problems. 23. Stability Theorems. 24. Stochastic Differential Equations. 25. Sturm-Liouville Theory. 26. Variational Equations. 27. Web Resources. 28. Well-Posed Differential Equations. 29. Wronskians & Fundamental Solutions. 30. Zeros of Solutions.

I.B. Transformations.

31. Canonical Forms. 32. Canonical Transformations. 33. Darboux Transformation. 34. An Involutory Transformation. 35. Liouville Transformation - 1. 36. Liouville Transformation - 2. 37. Changing Linear ODEs to a First Order System. 38. Transformations of Second Order Linear ODEs - 1. 39. Transformations of Second Order Linear ODEs - 2. 40. Transforming an ODE to an Integral Equation. 41. Miscellaneous ODE Transformations. 42. Transforming PDEs Generically. 43. Transformations of PDEs. 44. Transforming a PDE to a First Order System. 45. Pr fer Transformation. 46. Modified Pr fer Transformation. II. Exact Analytical Methods. 47. Introduction to Exact Analytical Methods. 48. Look-Up Technique. 49. Look-Up ODE Forms.

II.A Exact Methods for ODEs.

50. Use of the Adjoint Equation. 51. An Nth Order Equation. 52. Autonomous Equations - Independent Variable Missing. 53. Bernoulli Equation. 54. Clairaut's Equation. 55. Constant Coefficient Linear ODEs. 56 Contact Transformation. 57. Delay Equations. 58. Dependent Variable Missing. 59. Differentiation Method. 60. Differential Equations with Discontinuities. 61. Eigenfunction Expansions. 62. Equidimensional-in-x Equations. 63. Equidimensional-in-y Equations. 64. Euler Equations. 65. Exact First Order Equations. 66. Exact Second Order Equations. 67. Exact Nth Order Equations. 68. Factoring Equations. 69. Factoring/Composing Operators. 70. Factorization Method. 71. Fokker-Planck Equation. 72. Fractional Differential Equations. 73. Free Boundary Problems. 74. Generating Functions. 75. Green's Functions. 76. ODEs with Homogeneous Functions. 77. Hypergeometric Equation. 78. Method of Images. 79. Integrable Combinations. 80. Integrating Factors*. 81. Interchanging Dependent and Independent Variables. 82. Integral Representation: Laplace's Method. 83. Integral Transforms: Finite Intervals. 84. Integral Transforms: Infinite Intervals. 85. Lagrange's Equation. 86. Lie Algebra Technique. 87. Lie Groups: ODEs. 88. Non-normal Operators. 89. Operational Calculus. 90. Pfaffian Differential Equations. 91. Quasilinear Second Order ODEs. 92. Quasipolynomial ODEs. 93. Reduction of Order. 94. Resolvent Method for Matrix ODEs. 95. Riccati Equation - Matrices. 96. Riccati Equation - Scalars. 97. Scale Invariant Equations. 98. Separable Equations. 99. Series Solution. 100. Equations Solvable for x. 101. Equations Solvable for y. 102. Superposition. 103. Undetermined Coefficients. 104. Variation of Parameters. 105. Vector ODEs. II.B Exact Methods for PDEs. 106. B cklund Transformations. 107. Cagniard-de Hoop Method. 108. Method of Characteristics. 109. Characteristic Strip Equations. 110. Conformal Mappings. 111. Method of Descent. 112. Diagonalizable Linear Systems of PDEs. 113. Duhamel's Principle. 114. Exact Partial Differential Equations. 115. Fokas Method / Unified Transform. 116. Hodograph Transformation. 117. Inverse Scattering. 118. Jacobi's Method. 119. Legendre Transformation. 120. Lie Groups: PDEs. 121. Many Consistent PDEs. 122. Poisson Formula. 123. Resolvent Method for PDEs. 124. Riemann's Method 125 Separation of Variables. 126. Separable Equations: St ckel Matrix. 127. Similarity Methods. 128. Exact Solutions to the Wave Equation. 129. Wiener-Hopf Technique. III. Approximate Analytical Methods. 130. Introduction to Approximate Analysis. 131. Adomian Decomposition Method. 132. Chaplygin's Method. 133. Collocation. 134. Constrained Functions. 135. Differential Constraints. 136. Dominant Balance. 137. Equation Splitting. 138. Floquet Theory. 139. Graphical Analysis: The Phase Plane. 140 Graphical Analysis: Poincar