Differential Equations with Matlab

Brian Hunt and Ronald Lipsman are Professors Emeriti of Mathematics, and Jonathan Rosenberg is Ruth M. Davis Professor of Mathematics, at the University of Maryland.John E. Osborn was Professor Emeritus of Mathematics at the University of Maryland until his death in 2011.

• Preface v1 Introduction 11.1 Guiding Philosophy 11.2 Student’s Guide 31.3 Instructor’s Guide 51.3.1 MATLAB and Simulink 51.3.2 ODE Chapters 51.3.3 Computer Problem Sets 61.4 A Word about Software Versions 72 Getting Started with MATLAB 92.1 Platforms and Versions 92.2 Installation 102.3 Starting MATLAB 102.4 Typing in the Command Window 112.5 Online Help 112.6 MATLAB Windows 132.7 Ending a Session 143 Doing Mathematics with MATLAB 153.1 Arithmetic 153.2 Symbolic Computation 163.2.1 Substituting in Symbolic Expressions 173.2.2 Symbolic Expressions and Variable Precision Arithmetic 173.3 Vectors 183.3.1 Suppressing Output 193.4 Recovering from Problems 193.4.1 Errors in Input 203.4.2 Aborting Calculations 203.5 Functions 203.5.1 Built-in Functions 203.5.2 User-defined Functions 213.6 Managing Variables 213.7 Solving Equations 233.8 Graphics 253.8.1 Graphing with fplot 253.8.2 Modifying Graphs 263.8.3 Graphing with plot 263.8.4 Plotting Multiple Curves 283.8.5 Parametric Plots 283.8.6 Implicit Plots and Contour Plots 293.9 Calculus 313.10 Some Tips and Reminders 324 Using the Desktop and Scripts 334.1 The MATLAB Desktop 334.1.1 The Workspace 334.1.2 The Current Folder and Search Path 344.1.3 The Command History 354.2 Scripts and Functions 364.2.1 Plain Code Scripts 364.2.2 Live Scripts 384.2.3 Functions 394.3 Loops 404.4 Presenting Your Results 414.4.1 Presenting Graphics 424.4.2 Pretty Printing 444.4.3 “Publishing” a script 444.4.4 Preparing Homework Solutions 454.4.5 Exporting a Live Script 464.5 Debugging Your Scripts 48Problem Set A: Practice with MATLAB 515 Solutions of Differential Equations 555.1 Finding Symbolic Solutions 555.2 Existence and Uniqueness 585.3 Stability of Differential Equations 605.4 Different Types of Symbolic Solutions 636 Finer Points of the Symbolic Math Toolbox 697 A Qualitative Approach to Differential Equations 757.1 Direction Field for a First Order Linear Equation 757.2 Direction Field for a Non-Linear Equation 777.3 Autonomous Equations 797.3.1 Examples of Autonomous Equations 81Problem Set B: First Order Equations 858 Numerical Methods 978.1 Numerical Solutions Using MATLAB 988.2 Some Numerical Methods 1018.2.1 The Euler Method 1028.2.2 The Improved Euler Method 1058.2.3 The Runge-Kutta Method 1068.2.4 Inside ode45 1078.2.5 Round-off Error 1088.3 Controlling the Error in ode45 1088.4 Reliability of Numerical Methods 1099 Features of MATLAB 1139.1 Data Classes 1139.1.1 Symbolic and Floating Point Numbers 1149.1.2 Structures 1159.1.3 String Manipulation 1169.2 Functions and Expressions 1169.3 More about Scripts and Functions 1189.3.1 Variables and Input/Output in Scripts 1189.3.2 Variables in Functions 1189.3.3 Structure of Functions 1199.4 Matrices 1209.4.1 Solving Linear Systems 1219.4.2 Calculating Eigenvalues and Eigenvectors 1219.5 Graphics 1219.5.1 Figure Windows and Live Script Graphics 1229.5.2 Editing Figures 1239.6 Features of MATLAB’s Numerical ODE Solvers 1259.6.1 Evaluation of Numerical Solutions with deval 1259.6.2 Plotting Families of Numerical Solutions of ODEs 1269.6.3 Event Detection 1279.7 Troubleshooting 1299.7.1 The Most Common Mistakes 1299.7.2 Error and Warning Messages 13010 Using Simulink 13310.1 Constructing and Running a Simulink Model 13310.2 Output to the Workspace and How Simulink Works 138Problem Set C: Numerical Solutions 14311 Solving and Analyzing Second Order Linear Equations 15111.1 Second Order Equations with MATLAB 15311.2 Second Order Equations with Simulink 15711.3 Comparison Methods 15911.3.1 The Interlacing of Zeros 16011.3.2 Proof of the Sturm Comparison Theorem 16111.4 A Geometric Method 16211.4.1 The Constant Coefficient Case 16311.4.2 The Variable Coefficient Case 16411.4.3 Airy’s Equation 16511.4.4 Bessel’s Equation 16611.4.5 Other Equations 167Problem Set D: Second Order Equations 16912 Series Solutions 18312.1 Series Solutions 18412.2 Singular Points 18612.3 Other Linear and Nonlinear Equations 18713 Laplace Transforms 18913.1 Differential Equations and Laplace Transforms 19113.2 Discontinuous Functions 19413.3 Differential Equations with Discontinuous Forcing 196Problem Set E: Series Solutions and Laplace Transforms 19914 Higher Order Equations and Systems of First Order Equations 21314.1 Higher Order Linear Equations 21414.2 Systems of First Order Equations 21514.2.1 Linear First Order Systems 21514.2.2 Using MATLAB to Find Eigenpairs 21814.3 Phase Portraits 22214.3.1 Plotting a Single Trajectory 22214.3.2 Plotting Several Trajectories 22314.3.3 Numerical Solutions of First Order Systems 22514.3.4 A Non-Linear System 22715 Qualitative Theory for Systems of Differential Equations 229Problem Set F: Systems of Differential Equations 237Sample Solutions 255Index 276