# Introduction to Ordinary Differential Equations

### Academic Press International Edition

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Introduction to Ordinary Differential Equations is a 12-chapter text that describes useful elementary methods of finding solutions using ordinary differential equations. This book starts with an introduction to the properties and complex variable of linear differential equations. Considerable chapters covered topics that are of particular interest in applications, including Laplace transforms, eigenvalue problems, special functions, Fourier series, and boundary-value problems of mathematical physics. Other chapters are devoted to some topics that are not directly concerned with finding solutions, and that should be of interest to the mathematics major, such as the theorems about the existence and uniqueness of solutions. The final chapters discuss the stability of critical points of plane autonomous systems and the results about the existence of periodic solutions of nonlinear equations. This book is great use to mathematicians, physicists, and undergraduate students of engineering and the science who are interested in applications of differential equation.

## Table of Contents

Preface

1. 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

References

2. 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

References

3. Complex Variables

3.1 Introduction

3.2 Functions of a Complex Variable

3.3 Complex Series

3.4 Power Series

3.5 Taylor Series

References

4. 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

References

5. 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

References

6. 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

References

7. 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

References

8. 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

References

9. Systems of Differential Equations

9.1 First-Order Systems

9.2 Systems with Constant Coefficients

9.3 Applications

References

10. 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

References

11. 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

References

12. 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

References

Appendix

Answers to Miscellaneous Exercises

Subject Index

## Product details

- No. of pages: 444
- Language: English
- Copyright: © Academic Press 1966
- Published: January 1, 1966
- Imprint: Academic Press
- eBook ISBN: 9781483226224

## About the Author

### Albert L. Rabenstein

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