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Differential Equations with MATLAB
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Preface v

1 Introduction 1

1.1 Guiding Philosophy 1

1.2 Student's Guide 3

1.3 Instructor's Guide 5

1.3.2 ODE Chapters 5

1.3.3 Computer Problem Sets 6

1.4 A Word about Software Versions 7

2 Getting Started with MATLAB 9

2.1 Platforms and Versions 9

2.2 Installation 10

2.3 Starting MATLAB 10

2.4 Typing in the Command Window 11

2.6 MATLAB Windows 13

2.7 Ending a Session 14

3 Doing Mathematics with MATLAB 15

3.1 Arithmetic 15

3.2 Symbolic Computation 16

3.2.1 Substituting in Symbolic Expressions 17

3.2.2 Symbolic Expressions and Variable Precision Arithmetic 17

3.3 Vectors 18

3.3.1 Suppressing Output 19

3.4 Recovering from Problems 19

3.4.1 Errors in Input 20

3.4.2 Aborting Calculations 20

3.5 Functions 20

3.5.1 Built-in Functions 20

3.5.2 User-defined Functions 21

3.6 Managing Variables 21

3.7 Solving Equations 23

3.8 Graphics 25

3.8.1 Graphing with fplot 25

3.8.2 Modifying Graphs 26

3.8.3 Graphing with plot 26

3.8.4 Plotting Multiple Curves 28

3.8.5 Parametric Plots 28

3.8.6 Implicit Plots and Contour Plots 29

3.9 Calculus 31

3.10 Some Tips and Reminders 32

4 Using the Desktop and Scripts 33

4.1 The MATLAB Desktop 33

4.1.1 The Workspace 33

4.1.2 The Current Folder and Search Path 34

4.1.3 The Command History 35

4.2 Scripts and Functions 36

4.2.1 Plain Code Scripts 36

4.2.2 Live Scripts 38

4.2.3 Functions 39

4.3 Loops 40

4.4.1 Presenting Graphics 42

4.4.2 Pretty Printing 44

4.4.3 "Publishing" a script 44

4.4.4 Preparing Homework Solutions 45

4.4.5 Exporting a Live Script 46

Problem Set A: Practice with MATLAB 51

5 Solutions of Differential Equations 55

5.1 Finding Symbolic Solutions 55

5.2 Existence and Uniqueness 58

5.3 Stability of Differential Equations 60

5.4 Different Types of Symbolic Solutions 63

6 Finer Points of the Symbolic Math Toolbox 69 7 A Qualitative Approach to Differential Equations 75

7.1 Direction Field for a First Order Linear Equation 75

7.2 Direction Field for a Non-Linear Equation 77

7.3 Autonomous Equations 79

7.3.1 Examples of Autonomous Equations 81

Problem Set B: First Order Equations 85

8 Numerical Methods 97

8.1 Numerical Solutions Using MATLAB 98

8.2 Some Numerical Methods 101

8.2.1 The Euler Method 102

8.2.2 The Improved Euler Method 105

8.2.3 The Runge-Kutta Method 106

8.2.4 Inside ode45 107

8.2.5 Round-off Error 108

8.3 Controlling the Error in ode45 108

8.4 Reliability of Numerical Methods 109

9 Features of MATLAB 113

9.1 Data Classes 113

9.1.1 Symbolic and Floating Point Numbers 114

9.1.2 Structures 115

9.1.3 String Manipulation 116

9.2 Functions and Expressions 116

9.3 More about Scripts and Functions 118

9.3.1 Variables and Input/Output in Scripts 118

9.3.2 Variables in Functions 118

9.3.3 Structure of Functions 119

9.4 Matrices 120

9.4.1 Solving Linear Systems 121

9.4.2 Calculating Eigenvalues and Eigenvectors 121

9.5 Graphics 121

9.5.1 Figure Windows and Live Script Graphics 122

9.5.2 Editing Figures 123

9.6 Features of MATLAB's Numerical ODE Solvers 125

9.6.1 Evaluation of Numerical Solutions with deval 125

9.6.2 Plotting Families of Numerical Solutions of ODEs 126

9.6.3 Event Detection 127

9.7 Troubleshooting 129

9.7.1 The Most Common Mistakes 129

9.7.2 Error and Warning Messages 130

10.1 Constructing and Running a Simulink Model 133

10.2 Output to the Workspace and How Simulink Works 138

Problem Set C: Numerical Solutions 143 11 Solving and Analyzing Second Order Linear Equations 151

11.1 Second Order Equations with MATLAB 153

11.2 Second Order Equations with Simulink 157

11.3 Comparison Methods 159

11.3.1 The Interlacing of Zeros 160

11.3.2 Proof of the Sturm Comparison Theorem 161

11.4 A Geometric Method 162

11.4.1 The Constant Coefficient Case 163

11.4.2 The Variable Coefficient Case 164

11.4.3 Airy's Equation 165

11.4.4 Bessel's Equation 166

11.4.5 Other Equations 167

Problem Set D: Second Order Equations 169

12 Series Solutions 183

12.1 Series Solutions 184

12.2 Singular Points 186

12.3 Other Linear and Nonlinear Equations 187

13 Laplace Transforms 189

13.1 Differential Equations and Laplace Transforms 191

13.2 Discontinuous Functions 194

13.3 Differential Equations with Discontinuous Forcing 196

Problem Set E: Series Solutions and Laplace Transforms 199

14 Higher Order Equations and Systems of First Order Equations 213

14.1 Higher Order Linear Equations 214

14.2 Systems of First Order Equations 215

14.2.1 Linear First Order Systems 215

14.2.2 Using MATLAB to Find Eigenpairs 218

14.3 Phase Portraits 222

14.3.1 Plotting a Single Trajectory 222

14.3.2 Plotting Several Trajectories 223

14.3.3 Numerical Solutions of First Order Systems 225

14.3.4 A Non-Linear System 227

15 Qualitative Theory for Systems of Differential Equations 229

Problem Set F: Systems of Differential Equations 237

Sample Solutions 255

Index 276