Differential Equations with Matlab, 2nd

"Differential Equations with MATLAB" (2nd ed.) by B. Hunt, R. Lipsman, J.Osborn and J.Rosenberg is a supplemental text that can enrich and enhance any first course in ordinary differential equations. Designed to accompany Wiley's ODE texts written by Boyce/DiPrima, Borrelli/Coleman and Lomen/Lovelock, this supplement helps instructors move towards an earlier use of numerical and geometric methods, place a greater emphasis on systems (including nonlinear ones), and increase discussions of both the benefits and possible pitfalls in numerical solution of ODEs. By providing an introduction to the software that is integrated with the relevant mathematics, "Differential Equations with MATLAB" can perfectly complement and enhance these texts.

Preface

v

Introduction

1

(6)

Guiding Philosophy

1

(2)

Student's Guide

3

(1)

Instructor's Guide

4

(2)

Matlab and Simulink

5

(1)

ODE Chapters

5

(1)

Computer Problem Sets

5

(1)

A Word About Software Versions

6

(1)

Getting Started with Matlab

7

(6)

Platforms and Versions

7

(1)

Installation

8

(1)

Starting Matlab

8

(1)

Typing in the Command Window

9

(1)

Online Help

9

(2)

Matlab Windows

11

(1)

Ending a Session

12

(1)

Doing Mathematics with Matlab

13

(18)

Arithmetic

13

(1)

Recovering from Problems

14

(1)

Errors in Input

14

(1)

Aborting Calculations

14

(1)

Symbolic Computation

15

(2)

Substituting in Symbolic Expressions

15

(1)

Symbolic Expressions and Variable Precision Arithmetic

16

(1)

Vectors

17

(1)

Suppressing Output

18

(1)

Functions

18

(2)

Built-in Functions

18

(1)

User-defined Functions

19

(1)

Managing Variables

20

(1)

Solving Equations

21

(2)

Graphics

23

(6)

Graphing with ezplot

23

(1)

Modifying Graphs

24

(1)

Graphing with plot

25

(1)

Plotting Multiple Curves

26

(1)

Parametric Plots

26

(2)

Contour Plots

28

(1)

Calculus

29

(1)

Some Tips and Reminders

30

(1)

Using M-files and M-books

31

(16)

The Matlab Desktop

31

(3)

The Workspace

31

(1)

The Current Directory and Matlab Path

32

(1)

The Command History

33

(1)

M-files

34

(4)

Script M-files

34

(2)

Function M-files

36

(1)

Cells

37

(1)

Loops

38

(1)

Presenting Your Results

38

(7)

Presenting Graphics

39

(1)

Pretty Printing

40

(1)

``Publishing'' an M-file

41

(1)

M-books

41

(1)

Preparing Homework Solutions

42

(3)

Debugging Your M-files

45

(2)

Problem Set A: Practice with Matlab

47

(28)

Solutions of Differential Equations

51

(14)

Finding Symbolic Solutions

51

(3)

Existence and Uniqueness

54

(1)

Stability of Differential Equations

55

(4)

Different Types of Symbolic Solutions

59

(6)

A Qualitative Approach to Differential Equations

65

(10)

Direction Field for a First Order Linear Equation

65

(3)

Direction Field for a Non-Linear Equation

68

(1)

Autonomous Equations

69

(6)

Examples of Autonomous Equations

71

(4)

Problem Set B: First Order Equations

75

(56)

Numerical Methods

87

(16)

Numerical Solutions Using Matlab

88

(3)

Some Numerical Methods

91

(7)

The Euler Method

92

(3)

The Improved Euler Method

95

(1)

The Runge-Kutta Method

96

(1)

Inside ode45

97

(1)

Round-off Error

98

(1)

Controlling the Error in ode45

98

(1)

Reliability of Numerical Methods

99

(4)

Features of Matlab

103

(18)

Data Classes

103

(3)

String Manipulation

104

(1)

Symbolic and Floating Point Numbers

105

(1)

Structures

105

(1)

Functions and Expressions

106

(1)

More about M-files

107

(2)

Variables in Script M-files

108

(1)

Variables in Function M-files

108

(1)

Structure of Function M-files

109

(1)

Matrices

109

(2)

Solving Linear Systems

110

(1)

Calculating Eigenvalues and Eigenvectors

111

(1)

Graphics

111

(3)

Figure Windows

111

(2)

Editing Figures

113

(1)

Features of Matlab's Numerical ODE Solvers

114

(5)

Evaluation of Numerical Solutions with deval

115

(1)

Plotting Families of Numerical Solutions of ODEs

116

(1)

Event Detection

117

(2)

Troubleshooting

119

(2)

The Most Common Mistakes

119

(1)

Error and Warning Messages

120

(1)

Using Simulink

121

(10)

Constructing and Running a Simulink Model

121

(6)

Output to the Workspace and How Simulink Works

127

(4)

Problem Set C: Numerical Solutions

131

(26)

Solving and Analyzing Second Order Linear Equations

139

(18)

Second Order Equations with Matlab

141

(4)

Second Order Equations with Simulink

145

(2)

Comparison Methods

147

(3)

The Interlacing of Zeros

148

(1)

Proof of the Sturm Comparison Theorem

149

(1)

A Geometric Method

150

(7)

The Constant Coefficient Case

151

(1)

The Variable Coefficient Case

152

(1)

Airy's Equation

153

(1)

Bessel's Equation

154

(1)

Other Equations

155

(2)

Problem Set D: Second Order Equations

157

(36)

Series Solutions

171

(12)

Series Solutions

172

(2)

Singular Points

174

(3)

Function M-files for Series Solutions

177

(3)

Series Solutions Using maple

180

(3)

Laplace Transforms

183

(10)

Differential Equations and Laplace Transforms

185

(3)

Discontinuous Functions

188

(2)

Differential Equations with Discontinuous Forcing

190

(3)

Problem Set E: Series Solutions and Laplace Transforms

193

(38)

Higher Order Equations and Systems of First Order Equations

207

(16)

Higher Order Linear Equations

208

(1)

Systems of First Order Equations

209

(7)

Linear First Order Systems

209

(3)

Using Matlab to Find Eigenpairs

212

(4)

Phase Portraits

216

(7)

Plotting a Single Trajectory

216

(1)

Plotting Several Trajectories

217

(2)

Numerical Solutions of First Order Systems

219

(4)

Qualitative Theory for Systems of Differential Equations

223

(8)

Problem Set F: Systems of Differential Equations

231

(16)

Glossary

247

(12)

Sample Solutions

259

(26)

Index

285

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Differential Equations with Matlab, 2nd Edition

Summary

Table of Contents

Supplemental Materials