Preface© Copyright 2012 John Wiley and Sons, Inc.
Computers have at least three important uses in a differential equations course. The first is simply to crunch numbers, thereby generating accurate numerical approximations to solutions. The second is to carry out symbolic manipulations that would be tedious and time-consuming to do by hand. Finally, and perhaps most important of all, is the ability to translate the results of numerical or symbolic calculations into graphical form, so that the behavior of solutions can be easily visualized.
Boyce & DiPrima, Elementary Differential Equations, Tenth Edition, Preface.
Traditional introductory courses in ordinary differential equations (ODE) have concentrated on teaching a repertoire of techniques for finding formula solutions of various classes of differential equations. Typically, the result was rote application of formula techniques without a serious qualitative understanding of such fundamental aspects of the subject as stability, asymptotics, dependence on parameters, and numerical methods. These fundamental ideas are difficult to teach because they have a great deal of geometrical content and, especially in the case of numerical methods, involve a great deal of computation. Modern mathematical software systems, which are particularly effective for geometrical and numerical analysis, can help to overcome these difficulties. This book changes the emphasis in the traditional ODE course by using a mathematical software system to introduce numerical methods, geometric interpretation, symbolic computation, and qualitative analysis into the course in a basic way.
The mathematical software system we use is MATLAB®. (This book is also available in Mathematica and Maple versions.) We assume that the user has no prior experience with MATLAB. We include concise instructions for using MATLAB on four popular computer platforms: Windows, LINUX, Macintosh, and UNIX. This book is not a comprehensive introduction or reference manual to either MATLAB or any of the computer platforms. Instead, it focuses on the specific features of MATLAB that are useful for analyzing differential equations. We also discuss aspects of the MuPAD® language, used by MATLAB for symbolic computations, and of Simulink®, an auxiliary to MATLAB that is increasingly popular among scientists and engineers as a simulation tool. Wwe focus on the aspects of Simulink most useful for numerical and graphical solution of differential equations.
This supplement can easily be used in conjunction with most ODE textbooks. It addresses the standard topics in ordinary differential equations, but with a substantially different emphasis. We had two basic goals in mind when we introduced this supplement into our course. First, we wanted to deepen students' understanding of differential equations by giving them a new tool, a mathematical software system, for analyzing differential equations. Second, we wanted to bring students to a level of expertise in the mathematical software system that would allow them to use it in other mathematics, engineering, or science courses. We believe that we have achieved these goals in our own classes. We hope this supplement will be useful to students and instructors on other campuses in achieving the same goals.
AppreciationThe authors are deeply indebted to our late colleague John E. Osborn, who for more than two decades was an inspiration for all of us and for thousands of students. His expertise in differential equations and numerical methods, combined with his dedication to mathematics education, set a very high standard that we have tried our best to uphold. John was a guiding light in this long-term project and we sorely miss him.
Acknowledgment and DisclaimerWe are grateful for the essential contributions of our former colleagues Kevin R. Coombes and Garrett J. Stuck, who were co-authors of the first edition of this book.
We are pleased to acknowledge support of our research by the National Science Foundation, which contributed over many years to the writing of this book. Our work on the second edition was partially supported by NSF Grants 0616585, 0611094, and 0805003. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Brian R. Hunt
Ronald L. Lipsman
Jonathan M. Rosenberg
College Park, Maryland