## Ordinary Differential Equations Using MATLAB, 2/e

John Polking, Rice University
David Arnold, College of the Redwoods

Published May, 1999 by Prentice Hall Engineering/Science/Mathematics

Paper
ISBN 0-13-011381-6

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Differential Equations-Mathematics

A manual demonstrating the power and utility of MATLAB in a differential equations course. Accompanying the text is some of the finest software available for use in a differential equations course. This software has been extensively tested in the classroom and students applaud its ease of use. The MATLAB function dfield5 is used to plot solutions of first order differential equations of the form y'=f(t,y) using a variety of solvers: Euler, RK2, RK4, and Dormand-Prince. Direction fields can also be drawn. The MATLAB function pplane5 is used to draw solutions of planar, autonomous systems having the form x'=f(x,y), y'=g(x,y). Solutions can be drawn in the phase plane, tx-plane, ty-plane, and there are 3-dimensional and compositive views available. Many advanced features are available, including: direction field, nullclines, equilibrium point analysis and linearization, and the ability to adjust the step size of fixed-step solvers and the error tolerances of the Dormand-Prince solver. Users can also build and save systems and galleries of systems for later use. MATLAB functions eul, rk2, and rk4 also accompany the text and are used to introduce readers to fixed-step solvers and error analysis. An extensive treatment of MATLAB's suite of ODE solvers empowers students with the ability to apply powerful routines to the solution of higher order equations and systems. The text also features a honest treatment of the linear algebra needed for systems of ODEs, but no more.

NEW—The text is now compatible with MATLAB 5.
NEW—The text has been extensively rewritten. New exercises at various levels of difficulty have been added to aid a wider diversity of readers in their introduction to MATLAB 5.
NEW—A new section highlights the use of discontinuous forcing functions popular in engineering course.
NEW—Expanded treatment of the exponential matrix and its application to the solution of systems of differential equations.
NEW—A set of exercises designed to classify equilibrium points of linear systems in preparation for the analysis and linearization of nonlinear systems.

Preface.
1. Introduction to Matlab.

Numerical Expressions. Mathematical Functions. Output Format. Complex Arithmetic. Recording Your Work. Exercises.

2. Introduction to DFIELD5.

Starting DFIELD5. Initial Value Problems. Existence and Uniqueness. Qualitative Analysis. Zooming and Stopping. Using MATLAB While DFIELD5 is Open. Changing the Size and Appearance of the Display Window. Personalizing the Display Window. Printing, Quitting, and Using Clipboards. Exercises.

3. Vectors, Matrices, and Array Operations.

Matrices in MATLAB. Addition, Subtraction, and Scalar Multiplication. Vectors in MATLAB. Linear Combinations of Vectors. Matrix Multiplication. Array Operations. Plotting in MATLAB. Printing Your Plot. Simple Function M-files. Exercises.

4. Numerical Methods for ODEs.

Euler's Method. Euler's Method Versus the Exact Solution. Changing the Step Size---Script Files. Further Error Analysis. The Second Order Runge-Kutta Method. The Fourth Order Runge-Kutta Method. Comparing Euler, RK2, and RK4. MATLAB'S ODE45 Routine. Exercises. Appendix: M-files for Numerical Methods.

Step Functions. Step Functions in DFIELD5. Using Function M-files in DFIELD5. Solvers and Solver Settings in DFIELD5. Exercises. Appendix: SQW.M.

6. Introduction to PPLANE5.

Starting PPLANE5. Changing the Differential System---Using the PPLANE5 Setup Window. Plotting Solution Curves. Personalizing the Display Window. Saving Systems and Galleries. Exercises.

7. Solving ODEs in MATLAB.

MATLAB'S ODE Suite. Single First Order Differential Equations. Systems of First Order Equations. Second Order Differential Equations. The Lorenz System. Improving Accuracy. Kinky Plots. Behavior Near Discontinuities. Stiff Equations. Other Possibilities. Exercises. Student Projects.

8. The Symbolic Toolbox.

Symbolic Objects. The Default Symbolic Variable. Verifying Solutions to Differential Equations. Solving Ordinary Differential Equations. Solving Systems of Ordinary Differential Equations. Interpreting the Output of DSOLVE. The Solve command. Exercises.

9. Linear Algebra Using MATLAB.

Systems of Linear Equations. Matrix Indexing in MATLAB and Row Operations. The Rational Format. Solving Linear Equations. Determined Systems of Equations. The Determinant and Systems. The Nullspace of a Matrix. Linear Dependence and Independence. The Nullspace and Dependence. Exercises.

10. Homogeneous Linear Systems of ODEs.

Eigenvalues Using MATLAB. Eigenvectors Using MATLAB. MATLAB's EIG command. Tying It Together---Solving Systems. Complex Eigenvalues. The Exponential Matrix. Repeated Eigenvalues. Exercises.