# Differential Equations With MATLAB - GBV

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Differential Equationswith MATLAB (Third Edition)Updated for MATLAB 2011b (7.13),Simulink 7.8, and Symbolic Math Toolbox 5.7Brian R. HuntRonald L. LipsmanJohn E. OsbornJonathan M. RosenbergAll of the University of Maryland, College ParkWILEYJohn Wiley & Sons, Inc.New York Chichester Weinheim Brisbane Singapore Torontoin

ContentsPrefacev1Introduction1.1 Guiding Philosophy1.2 Student's Guide1.3 Instructor's Guide1.3.1 MATLAB, MuPAD, and Simulink1.3.2 ODE Chapters1.3.3 Computer Problem Sets1.4 A Word About Software Versions113555662Getting Started with MATLAB2.1 Platforms and Versions2.2 Installation2.3 Starting MATLAB2.4 Typing in the Command Window2.5 Online Help2.6 MATLAB Windows2.7 Ending a Session991010111113143Doing Mathematics with MATLAB3.1 Arithmetic3.2 Symbolic Computation3.2.1 Substituting in Symbolic Expressions3.2.2 Symbolic Expressions and Variable Precision Arithmetic3.3 Vectors3.3.1 Suppressing Output3.4 Recovering from Problems3.4.1 Errors in Input3.4.2 Aborting Calculations3.5 Functions3.5.1 Built-in Functions3.5.2 User-defined Functions3.6 Managing Variables1515161717181919202020202022Vll

viii4Contents3.73.8Solving EquationsGraphics3.8.1 Graphing with e z p l o t3.8.2 Modifying Graphs3.8.3 Graphing with p l o t3.8.4 Plotting Multiple Curves3.8.5 Parametric Plots3.8.6 Implicit Plots and Contour Plots3.9 Calculus3.10 Some Tips and Reminders23252526272828303132Using the Desktop and M-files4.1 The MATLAB Desktop4.1.1 The Workspace4.1.2 The Current Folder and Search Path4.1.3 The Command History4.2 M-files4.2.1 Script M-files4.2.2 Function M-files4.2.3 Cells4.3 Loops4.4 Presenting Your Results4.4.1 Presenting Graphics4.4.2 Pretty Printing4.4.3 "Publishing" an M-file4.4.4 M-books4.4.5 Preparing Homework Solutions4.5 Debugging Your M-files3333333435363638404041414343444547Problem Set A: Practice with MATLAB495Solutions of Differential Equations5.1 Finding Symbolic Solutions5.2 Existence and Uniqueness5.3 Stability of Differential Equations5.4 Different Types of Symbolic Solutions53535658616MuPAD697A Qualitative Approach to Differential Equations7.1 Direction Field for a First Order Linear Equation7.2 Direction Field for a Non-Linear Equation7.3 Autonomous Equations7.3.1 Examples of Autonomous Equations7575777981

ContentsixProblem Set B: First Order Equations858Numerical Methods8.1 Numerical Solutions Using MATLAB8.2 Some Numerical Methods8.2.1 The Euler Method8.2.2 The Improved Euler Method8.2.3 The Runge-Kutta Method8.2.4 Inside ode458.2.5 Round-off Error8.3 Controlling the Error in ode4 58.4 Reliability of Numerical Methods97981011021051061071081081099Features of MATLAB9.1 Data Classes9.1.1 String Manipulation9.1.2 Symbolic and Floating Point Numbers9.1.3 Structures9.2 Functions and Expressions9.3 More about M-files9.3.1 Variables in Script M-files9.3.2 Variables in Function M-files9.3.3 Structure of Function M-files9.4 Matrices9.4.1 Solving Linear Systems9.4.2 Calculating Eigenvalues and Eigenvectors9.5 Graphics9.5.1 Figure Windows9.5.2 Editing Figures9.6 Features ofMATLAB's Numerical ODE Solvers9.6.1 Evaluation of Numerical Solutions with d e v a l9.6.2 Plotting Families of Numerical Solutions of ODEs9.6.3 Event Detection9.7 Troubleshooting9.7.1 The Most Common Mistakes9.7.2 Error and Warning 12112312412512612712912912910 Using Simulink10.1 Constructing and Running a Simulink Model10.2 Output to the Workspace and How Simulink Works131131137Problem Set C: Numerical Solutions141

xContents11 Solving and Analyzing Second Order Linear Equations11.1 Second Order Equations with MATLAB11.2 Second Order Equations with Simulink11.3 Comparison Methods11.3.1 The Interlacing of Zeros11.3.2 Proof of the Sturm Comparison Theorem11.4 A Geometric Method11.4.1 The Constant Coefficient Case11.4.2 The Variable Coefficient Case11.4.3 Airy's Equation11.4.4 Bessel's Equation11.4.5 Other em Set D: Second Order Equations16712 Series Solutions12.1 Series Solutions12.2 Singular Points12.3 Other Linear and Nonlinear Equations18118218318513 Laplace Transforms13.1 Differential Equations and Laplace Transforms13.2 Discontinuous Functions13.3 Differential Equations with Discontinuous Forcing187189192194Problem Set E: Series Solutions and Laplace Transforms19714 Higher Order Equations and Systems of First Order Equations14.1 Higher Order Linear Equations14.2 Systems of First Order Equations14.2.1 Linear First Order Systems14.2.2 Using MATLAB to Find Eigenpairs14.3 Phase Portraits14.3.1 Plotting a Single Trajectory14.3.2 Plotting Several Trajectories14.3.3 Numerical Solutions of First Order Systems14.3.4 A Non-Linear System21121221321321622022022122322515 Qualitative Theory for Systems of Differential Equations227Problem Set F: Systems of Differential Equations235Sample Solutions253Index277

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

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