3 Doing Mathematics with MATLAB 13 4 Using M-files and M-books 31

4.1 The MATLAB Desktop 31

4.2 M-files 34

4.3 Loops 38

4.4 Presenting Your Results 38

4.5 Debugging Your M-files 45

Problem Set A: Practice with MATLAB 47 5 Solutions of Differential Equations 51

5.1 Finding Symbolic Solutions 51

5.2 Existence and Uniqueness 54

5.3 Stability of Differential Equations 55

5.4 Different Types of Symbolic Solutions 59

6 A Qualitative Approach to Differential Equations 65

6.1 Direction Field for a First Order Linear Equation 65

6.2 Direction Field for a Non-Linear Equation 68

6.3 Autonomous Equations 69

Problem Set B: First Order Equations 75 7 Numerical Methods 87

7.1 Numerical Solutions Using MATLAB 88

7.2 Some Numerical Methods 91

7.3 Controlling the Error in ode45 98

7.4 Reliability of Numerical Methods 99

8 Features of MATLAB 103

8.1 Data Classes 103

8.2 Functions and Expressions 106

8.3 More about M-files 107

8.4 Matrices 109

8.5 Graphics 111

8.6 Features of MATLAB 's Numerical ODE Solvers 114

8.7 Troubleshooting 119

9 Using Simulink 121

9.1 Constructing and Running a Simulink Model 121

9.2 Output to the Workspace and How Simulink Works 127

Problem Set C: Numerical Solutions 131 10 Solving and Analyzing Second Order Linear Equations 139

10.1 Second Order Equations with MATLAB 141

10.2 Second Order Equations with Simulink 145

10.3 Comparison Methods 147

10.4 A Geometric Method 150

Problem Set D: Second Order Equations 157 11 Series Solutions 171

11.1 Series Solutions 172

11.2 Singular Points 174

11.3 Function M-files for Series Solutions 177

11.4 Series Solutions Using maple 180

12 Laplace Transforms 183

12.1 Differential Equations and Laplace Transforms 185

12.2 Discontinuous Functions 188

12.3 Differential Equations with Discontinuous Forcing 190

Problem Set E: Series Solutions and Laplace Transforms 193 13 Higher Order Equations and Systems of First Order Equations 207

13.1 Higher Order Linear Equations 208

13.2 Systems of First Order Equations 209

13.3 Phase Portraits 216

14 Qualitative Theory for Systems of Differential Equations 223 Problem Set F: Systems of Differential Equations 231 Glossary 247 Sample Solutions 259 Index 285