A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, Chapman & Hall/CRC Press, 2006 To book contents

12. Ordinary Differential Equations

• 12.1. First-Order Differential Equations
  • 12.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems
  • 12.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration
  • 12.1.3. Exact Differential Equations. Integrating Factor
  • 12.1.4. Riccati Equation
  • 12.1.5. Abel Equations of the First Kind
  • 12.1.6. Abel Equations of the Second Kind
  • 12.1.7. Equations Not Solved for the Derivative
  • 12.1.8. Contact Transformations
  • 12.1.9. Approximate Analytic Methods for Solution of Equations
  • 12.1.10. Numerical Integration of Differential Equations
• 12.2. Second-Order Linear Differential Equations
  • 12.2.1. Formulas for the General Solution. Some Transformations
  • 12.2.2. Representation of Solutions as a Series in the Independent Variable
  • 12.2.3. Asymptotic Solutions
  • 12.2.4. Boundary Value Problems
  • 12.2.5. Eigenvalue Problems
  • 12.2.6. Theorems on Estimates and Zeros of Solutions
• 12.3. Second-Order Nonlinear Differential Equations
  • 12.3.1. Form of the General Solution. Cauchy Problem
  • 12.3.2. Equations Admitting Reduction of Order
  • 12.3.3. Methods of Regular Series Expansions with Respect to the Independent Variable
  • 12.3.4. Movable Singularities of Solutions of Ordinary Differential Equations. Painleve Transcendents
  • 12.3.5. Perturbation Methods of Mechanics and Physics
  • 12.3.6. Galerkin Method and Its Modifications (Projection Methods)
  • 12.3.7. Iteration and Numerical Methods
• 12.4. Linear Equations of Arbitrary Order
  • 12.4.1. Linear Equations with Constant Coefficients
  • 12.4.2. Linear Equations with Variable Coefficients
  • 12.4.3. Asymptotic Solutions of Linear Equations
  • 12.4.4. Collocation Method and Its Convergence
• 12.5. Nonlinear Equations of Arbitrary Order
  • 12.5.1. Structure of the General Solution. Cauchy Problem
  • 12.5.2. Equations Admitting Reduction of Order
• 12.6. Linear Systems of Ordinary Differential Equations
  • 12.6.1. Systems of Linear Constant-Coefficient Equations
  • 12.6.2. Systems of Linear Variable-Coefficient Equations
• 12.7. Nonlinear Systems of Ordinary Differential Equations
  • 12.7.1. Solutions and First Integrals. Uniqueness and Existence Theorems
  • 12.7.2. Integrable Combinations. Autonomous Systems of Equations
  • 12.7.3. Elements of Stability Theory
• References for Chapter 12

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