# Book:William E. Boyce/Elementary Differential Equations and Boundary Value Problems/Third Edition

## William E. Boyce and Richard C. DiPrima: Elementary Differential Equations and Boundary Value Problems (3rd Edition)

Published $\text {1977}$, Wiley

ISBN 0-471-83180-8

### Contents

Preface (William E. Boyce and William E. Boyce, Troy, New York, November $1976$)
Acknowledgments

1. INTRODUCTION
1.1 Classification of Differential Equations
1.2 Historical Remarks
2. FIRST ORDER DIFFERENTIAL EQUATIONS
2.1 Linear Equations
2.2 Further Discussion of Linear Equations
2.3 Nonlinear Equations
2.4 Separable Equations
2.5 Exact Equations
2.6 Integrating Factors
2.7 Homogeneous Equations
2.8 Miscellaneous Problems and Applications
2.9 Applications of First Order Equations
2.10 Elementary Mechanics
*2.11 The Existence and Uniqueness Theorem
Appendix. Derivation of Equation of Motion of a Body with Variable Mass

3. SECOND ORDER LINEAR EQUATIONS
3.1 Introduction
3.2 Fundamental Solutions of the Homogeneous Equation
3.3 Linear Independence
3.4 Reduction of Order
3.4 Complex Roots of the Characteristic Equation
3.5 Homogeneous Equations with Constant Coefficients
3.5.1 Complex Roots
3.6 The Nonhomogeneous Problem
3.6.1 The Method of Undetermined Coefficients
3.6.2 The Method of Variation of Parameters
3.7 Mechanical Vibrations
3.7.1 Free Vibrations
3.7.2 Forced Vibrations
3.8 Electrical Networks

4. SERIES SOLUTIONS OF SECOND ORDER LINEAR EQUATIONS
4.1 Introduction: Review of Power Series
4.2 Series Solutions Near an Ordinary Point, Part I
4.2.1 Series Solutions Near an Ordinary Point, Part II
4.3 Regular Singular Points
4.4 Euler Equations
4.5 Series Solutions Near a Regular Singular Point, Part I
4.5.1 Series Solutions Near a Regular Singular Point, Part II
*4.6 Series Solutions near a Regular Singular Point; $r_1 = r_2$ and $r_1 - r_2 = N$
*4.7 Bessel's Equation

5. HIGHER ORDER LINEAR EQUATIONS
5.1 Introduction
5.2 General Theory of $n$th Order Linear Equations
5.3 Homogeneous Equations with Constant Coefficients
5.4 The Method of Undetermined Coefficients
5.5 The Method of Variation of Parameters

6. THE LAPLACE TRANSFORM
6.1 Introduction. Definition of the Laplace Transform
6.2 Solution of Initial Value Problems
6.3 Step Functions
6.3.1 A Differential Equation with a Discontinuous Forcing Function
6.4 Impulse Functions
6.5 The Convolution Integral
6.6 General Discussion and Strategy

7. SYSTEMS OF FIRST ORDER LINEAR EQUATIONS
7.1 Introduction
7.2 Solution of Linear Systems by Elimination
7.3 Review of Matrices
7.4 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
7.5 Basic Theory of Systems of First Order Linear Equations
7.6 Homogeneous Linear Systems with Constant Coefficients
7.7 Complex Eigenvalues
7.8 Repeated Eigenvalues
7.9 Fundamental Matrices
7.10 Nonhomogeneous Linear Systems

8. NUMERICAL METHODS
8.1 Introduction
8.2 The Euler or Tangent Line Method
8.3 The Error
8.4 An Improved Euler Method
8.5 The Three-Term Taylor Series Method
8.6 The Runge-Kutta Method
8.7 Some Difficulties with Numerical Methods
8.8 A Multistep Method
8.9 Systems of First Order Equations

9. NONLINEAR DIFFERENTIAL EQUATIONS AND STABILITY
9.1 Introduction
9.2 Solutions of Autonomous Systems
9.3 The Phase Plane: Linear Systems
9.4 Stability; Almost Linear Systems
9.5 Competing Species and Predator-Prey Problems
9.6 Liapounov's Second Method
9.7 Periodic Solutions and Limit Cycles

10. PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES
10.1 Introduction
10.2 Heat Conduction and Separation of Variables
10.3 Fourier Series
10.4 The Fourier Theorem
10.5 Even and Odd Functions
10.6 Solution of Other Heat Conduction Problems
10.7 The Wave Equation: Vibrations of an Elastic String
10.8 Laplace's Equation
Appendix A. Derivation of the Heat Conduction Equation
Appendix B. Derivation of the Wave Equation

Chapter 11. BOUNDARY-VALUE PROBLEMS AND STURM-LIOUVILLE THEORY
11.1 Introduction
11.2 Linear Homogeneous Boundary Value Problems: Eigenvalues and Eigenfunctions
11.3 Sturm-Liouville Boundary Value Problems
11.4 Nonhomogeneous Boundary Value Problems
*11.5 Singular Sturm-Liouville Problems
*11.6 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion
*11.7 Series of Orthogonal Functions: Mean Convergence