Introduction to Ordinary Differential Equations

Preface1. Linear Differential Equations 1.1 Introduction 1.2 The Fundamental Theorem 1.3 First-Order Linear Equations 1.4 Linear Dependence 1.5 The Wronskian 1.6 Abel's Formula 1.7 Fundamental Sets of Solutions 1.8 Polynomial Operators 1.9 Equations with Constant Coefficients 1.10 Equations of Cauchy Type 1.11 The Nonhomogeneous Equation 1.12 Variation of Parameters 1.13 The Method of Undetermined Coefficients 1.14 Applications References2. Further Properties of Linear Differential Equations 2.1 Reduction of Order 2.2 Factorization of Operators 2.3 Some Variable Changes 2.4 Zeros of Solutions References3. Complex Variables 3.1 Introduction 3.2 Functions of a Complex Variable 3.3 Complex Series 3.4 Power Series 3.5 Taylor Series References4. Series Solutions 4.1 Introduction 4.2 Solutions at an Ordinary Point 4.3 Analyticity of Solutions at an Ordinary Point 4.4 Regular Singular Points 4.5 Solutions at a Regular Singular Point 4.6 The Method of Frobenius 4.7 The Case of Equal Exponents 4.8 When the Exponents Differ by a Positive Integer 4.9 Complex Exponents 4.10 The Point at Infinity References5. Bessel Functions 5.1 The Gamma Function 5.2 Bessel's Equation 5.3 Bessel Functions of the Second and Third Kinds 5.4 Properties of Bessel Functions 5.5 Modified Bessel Functions 5.6 Other Forms for Bessel's Functions References6. Orthogonal Polynomials 6.1 Orthogonal Functions 6.2 An Existence Theorem for Orthogonal Polynomials 6.3 Some Properties of Orthogonal Polynomials 6.4 Generating Functions 6.5 Legendre Polynomials 6.6 Properties of Legendre Polynomials 6.7 Orthogonality 6.8 Legendre's Differential Equation 6.9 Tchebycheff Polynomials 6.10 Other Sets of Orthogonal Polynomials References7. Eigenvalue Problems 7.1 Introduction 7.2 The Adjoint Equation 7.3 Boundary Operators 7.4 Self-Adjoint Eigenvalue Problems 7.5 Properties of Self-Adjoint Problems 7.6 Some Special Types of Self-Adjoint Problems 7.7 Singular Problems 7.8 Some Important Singular Problems References8. Fourier Series 8.1 Orthogonal Sets of Functions 8.2 Fourier Series 8.3 Examples of Fourier Series 8.4 Types of Convergence 8.5 Convergence in the Mean 8.6 Closed Orthogonal Sets 8.7 Pointwise Convergence of the Trigonometric Series 8.8 The Sine and Cosine Series 8.9 Other Fourier Series References9. Systems of Differential Equations 9.1 First-Order Systems 9.2 Systems with Constant Coefficients 9.3 Applications References10. Laplace Transforms 10.1 The Laplace Transform 10.2 Conditions for the Existence of the Laplace Transform 10.3 Properties of Laplace Transforms 10.4 Inverse Transforms 10.5 Application to Differential Equations References11. Partial Differential Equations and Boundary-Value Problems 11.1 Introduction 11.2 The Heat Equation 11.3 The Method of Separation of Variables 11.4 Steady State Heat Flow 11.5 The Vibrating String 11.6 The Solution of the Problem of the Vibrating String 11.7 The Laplacian in Other Coordinate Systems 11.8 A Problem in Cylindrical Coordinates 11.9 A Problem in Spherical Coordinates 11.10 Double Fourier Series References12. Nonlinear Differential Equations 12.1 First-Order Equations 12.2 Exact Equations 12.3 Some Special Types of Second-Order Equations 12.4 Existence and Uniqueness of Solutions 12.5 Existence and Uniqueness of Solutions for Systems 12.6 The Phase Plane 12.7 Critical Points 12.8 Stability for Nonlinear Systems 12.9 Perturbed Linear Systems 12.10 Periodic Solutions ReferencesAppendixAnswers to Miscellaneous ExercisesSubject Index