# A Modern Introduction to Differential Equations

A Modern Introduction to Differential Equations, Second Edition, provides an introduction to the basic concepts of differential equations. The book begins by introducing the basic concepts of differential equations, focusing on the analytical, graphical, and numerical aspects of first-order equations, including slope fields and phase lines. The discussions then cover methods of solving second-order homogeneous and nonhomogeneous linear equations with constant coefficients; systems of linear differential equations; the Laplace transform and its applications to the solution of differential equations and systems of differential equations; and systems of nonlinear equations. Each chapter concludes with a summary of the important concepts in the chapter. Figures and tables are provided within sections to help students visualize or summarize concepts. The book also includes examples and exercises drawn from biology, chemistry, and economics, as well as from traditional pure mathematics, physics, and engineering. This book is designed for undergraduate students majoring in mathematics, the natural sciences, and engineering. However, students in economics, business, and the social sciences with the necessary background will also find the text useful.

Audience
Undergraduate students studying differential equations.

Hardbound, 536 Pages

Published: March 2009

ISBN: 978-0-12-374746-4

## Contents

• Preface

Acknowledgements

Chapter 1 Introduction to Differential Equations

Introduction

1.1 Basic Terminology

1.2 Solutions of Differential Equations

1.3 Initial-Value Problems and Boundary-Value Problems

Summary

Project 1-1

Chapter 2 First-Order Differential Equations

Introduction

2.1 Separable Equations

2.2 Linear Equations

2.3 Compartment Problems

2.4 Slope Fields

2.5 Phase Lines and Phase Portraits

2.6 Equilibrium Points: Sinks, Sources, and Nodes

2.7 Bifurcations

2.8 Existence and Uniqueness of Solutions

Summary

Project 2-1

Project 2-2

Chapter 3 The Numerical Approximation of Solutions

Introduction

3.1 Eulerâs Method

3.2 The Improved Euler Method

3.3 More Sophisticated Numerical Methods: Runge-Kutta and Others

Summary

Project 3-1

Chapter 4 Second- and Higher-Order Equations

Introduction

4.1 Homogeneous Second-Order Linear Equations with Constant Coefficients

4.2 Nonhomogeneous Second-Order Linear Equations with Constant Coefficients

4.3 The Method of Undetermined Coefficients

4.4 Variation of Parameters

4.5 Higher-Order Linear Equations with Constant Coefficients

4.6 Higher-Order Equations and Their Equivalent Systems

4.7 The Qualitative Analysis of Autonomous Systems

4.8 Spring-Mass Problems

4.9 Existence and Uniqueness

4.10 Numerical Solutions

Summary

Project 4-1

Chapter 5 Systems of Linear Differential Equations

Introduction

5.1 Systems and Matrices

5.2 Two-Dimensional Systems of First-Order Linear Equations

5.3 The Stability of Homogeneous Linear Systems: Unequal Real Eigenvalues

5.4 The Stability of Homogeneous Linear Systems: Equal Real Eigenvalues

5.5 The Stability of Homogeneous Linear Systems: Complex Eigenvalues

5.6 Nonhomogeneous Systems

5.7 Generalizations: The n Ã n Case (n â¥ 3)

Summary

Project 5-1

Project 5-2

Chapter 6 The Laplace Transform

Introduction

6.1 The Laplace Transform of Some Important Functions

6.2 The Inverse Transform and the Convolution

6.3 Transforms of Discontinuous Functions

6.4 Transforms of Impulse Functions-The Dirac Delta Function

6.5 Transforms of Systems of Linear Differential Equations

6.6 A Qualitative Analysis via the Laplace Transform

Summary

Project 6-1

Chapter 7 Systems of Nonlinear Differential Equations

Introduction

7.1 Equilibria of Nonlinear Systems

7.2 Linear Approximation at Equilibrium Points

7.3 The PoincarÃ©-Lyapunov Theorem

7.4 Two Important Examples of Nonlinear Equations and Systems

7.5 Van Der Polâs Equation and Limit Cycles

Summary

Project 7-1

Appendix A Some Calculus Concepts and Results

A.1 Local Linearity: The Tangent Line Approximation

A.2 The Chain Rule

A.3 The Taylor Polynomial / Taylor Series

A.4 The Fundamental Theorem of Calculus (FTC)

A.5 Partial Fractions

A.6 Improper Integrals

A.7 Functions of Several Variables / Partial Derivatives

A.8 The Tangent Plane: The Taylor Expansion of F(x, y)

Appendix B Vectors and Matrices

B.1 Vectors and Vector Algebra; Polar Coordinates

B.2 Matrices and Basic Matrix Algebra

B.3 Linear Transformations and Matrix Multiplication

B.4 Eigenvalues and Eigenvectors

Appendix C Complex Numbers

C.1 Complex Numbers: The Algebraic View

C.2 Complex Numbers: The Geometric View

C.3 The Quadratic Formula

C.4 Eulerâs Formula

Appendix D Series Solutions of Differential Equations

D.1 Power Series Solutions of First-Order Equations

D.2 Series Solutions of Second-Order Linear Equations: Ordinary Points

D.3 Regular Singular Points: The Method of Frobenius

D.4 The Point at Infinity

D.5 Some Additional Special Differential Equations

Answers/Hints to Odd-Numbered Exercises

Index