Solutions Manual Introduction Differential

Solutions Manual to

Introduction to Differential Equations with Dynamical Systems

by Stephen L. Campbell and Richard Haberman

M. Ziaul Haque

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

Copyright c 2008 by Princeton University Press

Published by Princeton University Press 41 William Street, Princeton, New Jersey 08540

In the United Kingdom: Princeton University Press 6 Oxford Street, Woodstock, Oxfordshire, 0X20 1TW

All Rights Reserved

This book has been composed in LATEX press.princeton.edu

Contents

Preface

v

Chapter 1. First-Order Differential Equations and Their Applications

1

1.1 Introduction to Ordinary Differential Equations

1

1.2 Definite Integral and the Initial Value Problem

1

1.3 First-Order Separable Differential Equations

3

1.4 Direction Fields

5

1.5 Euler's Numerical Method (Optional)

7

1.6 First-Order Linear Differential Equations

10

1.7 Linear First-Order Differential Equations with Constant Coeffi

cients and Constant Input

15

1.8 Growth and Decay Problems

20

1.9 Mixture Problems

23

1.10 Electronic Circuits

25

1.11 Mechanics II: Including Air Resistance

26

1.12 Orthogonal Trajectories (optional)

27

Chapter 2. Linear Second and Higher-Order Differenial Equations

29

2.1 General Solution of Second-Order Linear Differential Equations

29

2.2 Initial Value Problem (For Homogeneous Equation)

30

2.3 Reduction of Order

32

2.4 Homogeneous Linear Constant Coefficient Differential Equations

(Second Order)

35

2.5 Mechanical Vibrations I: Formulation and Free Response

39

2.6 The Method of Undetermined Coefficients

45

2.7 Mechanical Vibrations II: Forced Response

58

2.8 Linear Electric Circuits

65

2.9 Euler Equation

68

2.10 Variation of Parameters (Second-Order)

70

2.11 Variation of Parameters (nth-Order)

75

Chapter 3. The Laplace Transform

82

3.1 Definition and Basic Properties

82

3.2 Inverse Laplace Transforms (Roots, Quadratics, & Partial Fractions) 86

3.3 Initial-Value Problems for Differential Equations

94

3.4 Discontinuous Forcing Functions

98

3.5 Periodic Functions

109

3.6 Integrals and the Convolution Theorem

114

3.7 Impulses and Distributions

118

iv

CONTENTS

Chapter 4. An Introduction to Linear Systems of Differential Equations and

Their Phase Plane

121

4.1 Introduction

121

4.2 Introduction to Linear Systems of Differential Equations

121

4.3 Phase Plane for Linear Systems of Differential Equations

130

Chapter 5. Mostly Nonlinear First-Order Differential Equations

142

5.1 First-Order Differential Equations

142

5.2 Equilibria and Stability

142

5.3 One Dimensional Phase Lines

143

5.4 Application to Population Dynamics: The Logistic Equation

146

Chapter 6. Nonlinear Systems of Differential Equations in the Plane

150

6.1 Introduction

150

6.2 Equilibria of Nonlinear Systems, Linear Stability Analysis of Equi

librium, and Phase Plane

150

6.3 Population Models

161

6.4 Mechanical Systems

178

Preface

This Student Solutions Manual contains solutions to the odd-numbered ex ercises in the text Introduction to Differential Equations with Dynamical Systems by Stephen L. Campbell and Richard Haberman.

To master the concepts in a mathematics text the students must solve prob lems which sometimes may be challenging. This manual has been written focusing student's needs and expectations. Instead of providing only the answer with very few steps, I include a reasonably detailed solution with a fair amount of detail when explaining the solution of the problem. The solutions are self-explanatory and consistent with the notations and termi nologies used in the text book. I hope this manual will help students build problem-solving skills.

I would like to thank many people who have provided invaluable help, in many ways, in the preparation of this manual. First, I take this opportunity to thank Professor Richard Haberman for his generous expert help, construc tive comments and accuracy checking. I would also like to thank Professor Stephen L. Campbell for assembling the final manuscript, Professor Peter K. Moore for facilitating support process and Ms. Vickie Kearn of the pub lishing company for her patience and support. Finally, I must appreciate the patience of my wife, Rukshana, and my daughters, Zareen and Ehram for their understanding and compromise of summer time that was slighted because of my busy schedule.

M. Ziaul Haque

Southern Methodist University

Dallas, TX, 75275, U.S.A.

July, 2007.

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