# Introduction to Ordinary Differential Equations

### Second Enlarged Edition with Applications

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Introduction to Ordinary Differential Equations, Second Edition provides an introduction to differential equations. This book presents the application and includes problems in chemistry, biology, economics, mechanics, and electric circuits. Organized into 12 chapters, this edition begins with an overview of the methods for solving single differential equations. This text then describes the important basic properties of solutions of linear differential equations and explains higher-order linear equations. Other chapters consider the possibility of representing the solutions of certain linear differential equations in terms of power series. This book discusses as well the important properties of the gamma function and explains the stability of solutions and the existence of periodic solutions. The final chapter deals with the method for the construction of a solution of the integral equation and explains how to establish the existence of a solution of the initial value system. This book is a valuable resource for mathematicians, students, and research workers.

## Table of Contents

Preface

I Introduction to Differential Equations

1.1 Introduction

1.2 Separable Equations

1.3 Exact Equations

1.4 First-Order Linear Equations

1.5 Orthogonal Trajectories

1.6 Decay and Mixing Problems

1.7 Population Growth

1.8 An Economic Model

1.9 Cooling: The Rate of a Chemical Reaction

1.10 Two Special Types of Second-Order Equations

1.11 Falling Bodies

References

II Linear Differential Equations

2.1 Introduction

2.2 Linear Dependence

2.3 Wronskians

2.4 Polynomial Operators

2.5 Complex Solutions

2.6 Equations with Constant Coefficients

2.7 Cauchy-Euler Equations

2.8 Nonhomogeneous Equations

2.9 The Method of Undetermined Coefficients

2.10 Variation of Parameters

2.11 Simple Harmonic Motion

2.12 Electric Circuits

2.13 Theory of Linear Equations

References

III Series Solutions

3.1 Power Series

3.2 Taylor Series

3.3 Ordinary Points

3.4 Regular Singular Points

3.5 The Case of Equal Exponents

3.6 The Case When the Exponents Differ by an Integer

3.7 The Point at Infinity

3.8 Convergence of the Series

References

IV Bessel Functions

4.1 The Gamma Function

4.2 Bessel's Equation

4.3 Bessel Functions of the Second and Third Kinds

4.4 Properties of Bessel Functions

4.5 Modified Bessel Functions

4.6 Other Forms for Bessel's Equation

References

V Orthogonal Polynomials

5.1 Orthogonal Functions

5.2 An Existence Theorem for Orthogonal Polynomials

5.3 Properties of Orthogonal Polynomials

5.4 Generating Functions

5.5 Legendre Polynomials

5.6 Properties of Legendre Polynomials

5.7 Orthogonality

5.8 Legendre's Differential Equation

5.9 Tchebycheff Polynomials

5.10 Other Sets of Orthogonal Polynomials

Table of Orthogonal Polynomials

References

VI Eigenvalue Problems

6.1 Introduction

6.2 Self-Adjoint Problems

6.3 Some Special Cases

6.4 Singular Problems

6.5 Some Important Singular Problems

References

VII Fourier Series

7.1 Introduction

7.2 Examples of Fourier Series

7.3 Types of Convergence

7.4 Convergence in the Mean

7.5 Complete Orthogonal Sets

7.6 Pointwise Convergence of the Trigonometric Series

7.7 The Sine and Cosine Series

7.8 Other Fourier Series

References

VIII Systems of Differential Equations

8.1 Introduction

8.2 First-Order Systems

8.3 Linear Systems with Constant Coefficients

8.4 Mechanical Systems

8.5 Electric Circuits

8.6 Some Problems from Biology

References

IX Laplace Transforms

9.1 The Laplace Transform

9.2 Conditions for the Existence of the Laplace Transform

9.3 Properties of Laplace Transforms

9.4 Inverse Transforms

9.5 Applications to Differential Equations

References

X Partial Differential Equations and Boundary Value Problems

10.1 Introduction

10.2 The Heat Equation

10.3 The Method of Separation of Variables

10.4 Steady-State Heat Flow

10.5 The Vibrating String

10.6 The Solution of the Problem of the Vibrating String

10.7 The Laplacian in Other Coordinate Systems

10.8 A Problem in Cylindrical Coordinates

10.9 A Problem in Spherical Coordinates

10.10 Double Fourier Series

References

XI The Phase Plane

11.1 Introduction

11.2 Stability

11.3 The Method of Liapunov

11.4 Perturbed Linear Systems

11.5 Periodic Solutions

References

XII Existence and Uniqueness of Solutions

12.1 Preliminaries

12.2 Successive Approximations

12.3 Vector Functions

12.4 First-Order Systems

References

Appendix

A1 Determinants

A2 Properties of Determinants

A3 Cofactors

A4 Cramer's Rule

Answers to Selected Exercises

Subject Index

## Product details

- No. of pages: 536
- Language: English
- Copyright: © Academic Press 1972
- Published: January 1, 1972
- Imprint: Academic Press
- eBook ISBN: 9781483263854

## About the Author

### Albert L. Rabenstein

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