Preface iii
1 Introduction 1
1.1 Guiding Philosophy 1
1.2 Student's Guide 3
1.3 Instructor's Guide 5
1.4 A Word About Software Versions 6
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
3.3 Vectors 18
3.4 Recovering from Problems 19
3.5 Functions 20
3.6 Managing Variables 22
3.7 Solving Equations 23
3.8 Graphics 25
3.9 Calculus 31
3.10 Some Tips and Reminders 32
4 Using the Desktop and M-files 33
4.1 The MATLAB Desktop 33
4.2 M-files 36
4.3 Loops 40
Problem Set A: Practice with MATLAB 47
5 Solutions of Differential Equations 53
5.1 Finding Symbolic Solutions 53
5.2 Existence and Uniqueness 56
5.3 Stability of Differential Equations 58
5.4 Different Types of Symbolic Solutions 61
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
Problem Set B: First Order Equations 85
8 Numerical Methods 97
8.1 Numerical Solutions Using MATLAB 98
8.2 Some Numerical Methods 101
8.3 Controlling the Error in ode45 108
8.4 Reliability of Numerical Methods 109
9 Features of MATLAB 113
9.1 Data Classes 113
9.2 Functions and Expressions 116
9.4 Matrices 119
9.5 Graphics 121
9.6 Features of MATLAB's Numerical ODE Solvers 124
9.7 Troubleshooting 129
10.1 Constructing and Running a Simulink Model 131
10.2 Output to the Workspace and How Simulink Works 137
Problem Set C: Numerical Solutions 141
11 Solving and Analyzing Second Order Linear Equations 149
11.1 Second Order Equations with MATLAB 151
11.2 Second Order Equations with Simulink 155
11.3 Comparison Methods 157
11.4 A Geometric Method 160
Problem Set D: Second Order Equations 167
12 Series Solutions 181
12.1 Series Solutions 182
12.2 Singular Points 183
12.3 Other Linear and Nonlinear Equations 185
13 Laplace Transforms 187
13.1 Differential Equations and Laplace Transforms 189
13.2 Discontinuous Functions 192
13.3 Differential Equations with Discontinuous Forcing 194
Problem Set E: Series Solutions and Laplace Transforms 197
14 Higher Order Equations and Systems of First Order Equations 211
14.1 Higher Order Linear Equations 212
14.2 Systems of First Order Equations 213
14.3 Phase Portraits 220
15 Qualitative Theory for Systems of Differential Equations 227
Problem Set F: Systems of Differential Equations 235
Sample Solutions 253
Index 277