In recent years, partially due to evolving technology and its expanding importance in pedagogy and to the reform movement in calculus, changes have occurred in the course called "Differential Equations." Instructors are questioning aspects of both the traditional teaching methods used and the traditional content of the course. This healthy introspection is important in making the subject matter not only more interesting for students but also more relevant to the world in which they live.

PREFACE

ACKNOWLEDGMENTS

1 INTRODUCTION TO DIFFERENTIAL EQUATIONS

　1.1 Definitions and Terminology

　1.2 Initial-Value Problems

　1.3 Differential Equations as Mathematical Models

　Chapter 1 in Review

2 FIRST-ORDER DIFFERENTIAL EQUATIONS

　2.1 Solution Curves Without the Solution

　2.2 Separable Variables

　2.3 Linear Equations

　2.4 Exact Equations

　2.5 Solutions by Substitutions

　2.6 A Numerical Solution

　Chapter 2 in Review

3 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS

　3.1 Linear Equations

　3.2 Nonlinear Equations

　3.3 Systems of Linear and Nonlinear Differential Equations

　Chapter 3 in Review

　Project Module: Harvesting of Renewable Natural Resources, by Gilbert N. Lewis

4 HIGHER-ORDER DIFFERENTIAL EQUATIONS

　4.1 Preliminary Theory: Linear Equations

　　4.1.1 Initial-Value and Boundary-Value Problems

　　4.1.2 Homogeneous Equations

　　4.1.3 Nonhomogeneous Equations

　4.2 Reduction of Order

　4.3 Homogeneous Linear Equations with Constant Coefficients

　4.4 Undetermined Coefficients--Superposition Approach

　4.5 Undetermined Coefficients--Annihilator Approach

　4.6 Variation of Parameters

　4.7 Cauchy-Euler Equation

　4.8 Solving Systems of Linear Equations by Elimination

　4.9 Nonlinear Equations

　Chapter 4 in Review

5 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS

　5.1 Linear Equations: Initial-Value Problems

　　5.1.1 Spring/Mass Systems: Free Undamped Motion

　　5.1.2 Spring/Mass Systems: Free Damped Motion

　　5.1.3 Spring/Mass Systems: Driven Motion

　　5.1.4 Series Circuit Analogue

　5.2 Linear Equations: Boundary-Value Problems

　5.3 Nonlinear Equations

 Chapter 5 in Review

 Project Module: The Collapse of the Tacoma Narrows Suspension Bridge, by Gilbert N. Lewis

6 SERIES SOLUTIONS Of LINEAR EQUATIONS

 6.1 Solutions About Ordinary Points

6.1.1 Review of Power Series

6.1.2 Power Series Solutions

 6.2 Solutions About Singular Points

 6.3 Two Special Equations

 Chapter 6 in Review

7 THE LAPLACE TRANSFORM

　7.1 Definition of the Laplace Transform

　7.2 Inverse Transform and Transforms of Derivatives

　7.3 Translation Theorems

　　7.3.1 Translation on the s-Axis

　　7.3.2 Translation on the t-Axis

　7.4 Additional Operational Properties

　7.5 Dirac Delta Function

　7.6 Systems of Linear Equations

　Chapter 7 in Review

8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS

　8.1 Preliminary Theory

　　8.2 Homogeneous Linear Systems with Constant Coefficients

　　8.2.1 Distinct Real Eigenvalues

　　8.2.2 Repeated Eigenvalues

　　8.2.3 Complex Eigenvalues

　8.3 Variation of Parameters

　8.4 Matrix Exponential

　Chapter 8 in Review

　Project Module: Earthquake Shaking of Multistory Buildings, by Gilbert N. Lewis

9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

 9.1 Euler Methods and Error Analysis

 9.2 Runge-KuttaMothods

 9.3 MultistepMethods

 9.4 Higher-Order Equations and Systems

 9.5 Second-Order Boundary-Value Problems

　Chapter 9 in Review

APPENDIXES APP-1

Ⅰ Gramma Functions APP-1

Ⅱ Introduction to Matrices APP-3

Ⅲ Laplace Transforms APP-25

SELECTED ANSWERS FOR ODD-NUMBERED PROBLEMS AN-1

INDEX I-1