Gray

Mezzino

Pinsky

Preface of the Text


Our Perspective

 The purpose of this book is to provide a traditional treatment of elementary ordinary differential equations while introducing the computer-assisted methods that are now available with Mathematica. We have chosen Mathematica over other systems of computer algebra because of its combination of easy access and computational power, as evidenced through symbolic, numerical, and graphical output.

In order to make this work totally self-contained, we have developed the fundamentals of differential equations from the very beginning. This includes the solution methods for the traditional classes of solvable equations (first-order linear, second-order constant-coefficients, linear systems, Laplace transforms, power series solutions and so forth) as well as a presentation of the basic theory of existence/uniqueness and the traditional numerical methods for first-order equations. In the process some new mathematical points have been developed, to be noted below. It is our firm belief that a solid mastery of the subject of differential equations can only be achieved through a strong traditional course. This is enhanced by the graphical capabilities of Mathematica, which have allowed the incorporation of many more graphs than are normally available in books at this level.

When we move beyond the traditional course and envisage the integration of computers, many potential problems arise in redesigning the syllabus. A well-known first attempt was the development of BASIC and similar programs in the 1970s. The ease with which numerical routines such as the Euler method could be implemented made the new software a very useful pedagogical device. However, computer technology distracted students from learning about differential equations.

The scene changed dramatically with the advent of symbolic manipulation programs---first Macsyma, then Derive, Maple, Mathematica and others---which made symbolic as well as numerical solutions of differential equations possible via computers. In principle symbolic manipulation programs can perform routine calculations, permitting students to cover more theory and applications. In practice, it may be necessary for students to learn a fair amount about a symbolic manipulation program for a course in differential equations. Certainly, using Mathematica is far simpler than writing programs in C or BASIC. Nevertheless, students and professors frequently become frustrated when Mathematica does not behave exactly as mathematics does, and valuable time is wasted. It is for this reason that our text provides a parallel development of classical solution methods as well as a special Mathematica package, ODE.m. No prior knowledge of Mathematica is required either to use this book or the programs contained in ODE.m.

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Prerequisites

 We presume knowledge of calculus, but a full course in linear algebra is not required. Vectors and matrices are not needed until Chapter 15; students can easily learn the necessary material from the linear algebra review provided in Appendix A.

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Features for Instructors
  • Power series. In Chapters 20 and 21 we present a new and efficient method for obtaining the power series solution or Frobenius solution to a differential equation. The method uses an algorithm that goes back to Cauchy. (The standard method is also explained.)

  • Phase portraits. We have also achieved some coherence in the discussion of two-dimensional linear systems with constant-coefficients. Thirteen separate cases are enumerated and discussed in detail in Chapter 16.

  • Laplace transforms. Chapter 14 contains a detailed treatment of Laplace transforms because of their importance in engineering. Discontinuous forcing functions and the Dirac delta function are fully treated. Furthermore, ODE automates the use of Laplace transforms, much more so than is done with Mathematica's built-in packages.

  • Seminumerical solutions. Since it is usually impossible to find the general solution of a constant-coefficient linear differential equation whose order is greater than 4, we show how to find numerical approximations to the roots of the associated characteristic equation and to use them to construct an approximate general solution.

  • Mathematical details. Proofs of many important theorems are given in optional subsections. A criterion for global existence of first-order nonlinear equations is presented as a complement to the classical Picard local existence proof in Chapter 5.

  • Other solution methods. For historical completeness we have included the solution methods of Bernoulli, Clairaut, and Lagrange. Since these are rarely used in practice, these methods can be skipped in all but the most extensive differential equations courses. However, since they are available in ODE.m, they can be invoked at a moment's notice on those infrequent occasions when they are needed.
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Features for Students
  • Mathematica. With some experience, it will be found that the use of Mathematica, and in particular ODE.m, will simplify a student's ability to deal with differential equations. The principal command ODE is as easy to use as a hand calculator, but much more powerful. While we emphasize the importance of mastering the traditional methods, it is no longer necessary to continually suffer (as students did 50 years ago) doing lengthy hand calculations to solve differential equations.

  • Exercises and worked examples. Most people learn at least in part by imitation and repetition. Accordingly, we have included more than 300 worked examples in the text and more than 650 exercises. Past experience has shown that this method will guarantee efficient mastery of the traditional methods of solution. There are also many worked examples, exercises, and solutions related to the Mathematica material. All of these will be included in the CD-ROM with the book. Students can have practice solving differential equations, with and without Mathematica/.

  • Procedure boxes. Important solution techniques are summarized in boxes; these convenient summaries are easy to find.

  • Computer graphics. All graphics in this book have been generated by Mathematica; this is a guarantee of their accuracy. When writing this book, we have found time after time that a graph adds a new dimension to understanding a problem, much more so than either a formula or a table of values. The CD-ROM also includes color graphics.

  • Historical footnotes. The history of differential equations is both fascinating and important. Historical footnotes, including photo portraits of these individuals, provide details about some of the scientists involved in its development.

  • Extensive indices. There are three indices: a general index, a name index, and a Mathematica index. The Mathematica index can be used by those who already know Mathematica to find examples of how various Mathematica commands are used.
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Features for Programmers

 Although the package ODE.m can be used to mask many details of the solving of differential equations, we provide in optional sections details on how ODE.m does its job. In most cases this involves translating certain mathematical formulas to the formalism of Mathematica. Complete descriptions of the miniprograms that make up ODE.m are provided on the CD-ROM. These easy-to-understand miniprograms provide valuable insight not only on how Mathematica goes about its job, but also about the solution process for many types of ordinary differential equations.

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Connections with the Traditional Course

 This book can be used as a traditional text, with no reference to Mathematica or computers. In every case, the computer enhancement always follows the traditional text. The following points are noted for reference:

  • The mathematical theory of first-order differential equations is given in Chapter 3, and the implementation using ODE.m is given in Chapter 4.

  • The mathematical theory of second-order linear differential equations is given in Chapters 8 and 9, and the implementation using ODE.m is given in Chapter 10.

  • The mathematical theory of higher-order linear differential equations is given in Sections 12.1 --12.3, and the implementation using ODE.m is given in Sections 12.4 --12.5.

  • The mathematical theory of numerical solutions to differential equations is given in Sections 13.1 --13.3, and the implementation using ODE.m is given in Sections13.4 --13.8.

  • The mathematical theory of Laplace transforms is given in Sections 14.1 --14.9, and the implementation using ODE.m is given in Section 14.10.

  • The mathematical theory of linear systems is given in Chapter 15, and the implementation using ODE.m is given in Chapter 16.

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Use of the CD-ROM

 Together with this book, we have developed a CD-ROM that allows the user to have unprecedented access to aspects of differential equations, that were previously unavailable with conventional texts. This allows the instructor and student to have easy access to the symbolic, graphical, and dynamic aspects of the subject. In detail, the CD-ROM has a number of directories. For your convenience, the following links will take you to the platform independent Mathematica 3.0 notebooks and related files for each of the following sections:

  • The Mathematica solution of worked examples: In order to access the Mathematica notebook containing Example 5.10, one opens Chapter_05.ma and then searches for Example 5.10.

  • The Mathematica solution of exercises: In order to find the solution of Exercise 3 in Section 14.5, one selects the subdirectory Chapter 14, then opens Sec1405.ma and searches for Exercise 3.

  • Sample Labs: For the student who wants extra practice, we have organized seven laboratory assignments using Mathematica. The entire collection gives a representative sampling of the entire book. Each assignment opens with a review of the relevant theory, followed by a sample problem preceding the new exercise. The topics include first-order equations, applications to cell growth, second-order constant-coefficient equations, numerical solutions, Laplace transforms, series solutions, and linear systems.

  • Miscellaneous Mathematica notebooks: Among the additional features available in this directory we have a comparison of different techniques of numerical integration, various direction fields of two-dimensional systems, two and three-dimensional phase portraits, a tour of highlights of the ODE.m package, phase portraits and other aspects of the pendulum, and examples of resonance using Mathematica's Play function.

  • Mathematica movies: The relevant subdirectories are (i) LinearSystems, (ii) NonlinearSystems, (iii) SeriesApproximations, (iv) VectorFields. These cover, respectively, 2D and 3D phase portraits related to Chapter 16; springs, pendula, Lorentz attractors, VanderPol attractors, and predator-prey models; the Picard series approximation to the solution of y' = t2 - y , y(1) = 2 on the interval -2 < t < 6; a Mathematica movie of the flow corresponding to the equation y' = sin(t)/y on a rectangle centered at (0,0).

  • Packages: This directory contains ODE.m, which includes the Mathematica functions that are frequently used throughout the book. Other Mathematica functions are found in the auxiliary package ODEx.m.

  • ODE Reference Manual: This contains the Manual which documents the Mathematica code in ODE.m. It can be used as a beginners handbook and quick reference manual for ordinary differential equations. Although this manual can be read with any HTML browser, it is also present in encapsulated postscript form in the ``psrefman'' folder for those with a postscript viewer or postscript printer.

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Support for Different Versions of Mathematica

 As we go to press, Wolfram Research Inc. is announcing the release of Mathematica 3.0, as documented in The Mathematica Book, by Stephen Wolfram, Third Edition, Cambridge University Press, 1996. All of the programs in our book and in ODE.m work both in Version 2.2 and Version 3.0 of Mathematica. In addition, we are developing some specific new notebooks that are uniquely adapted to Mathematica 3.0 that are available on the Web site version of the CD-ROM, which is accessible through http://math.cl.uh.edu/ode/ode.html.

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Supplements and Software
  • Solutions manual. Short answers to the odd-numbered problems are given at the end of the text. Complete solutions are given in a separate solutions manual.

  • ODE.m. This package is included with the book and is also available via anonymous ftp.

  • Notebooks. Mathematica notebooks, both for Version 2.2 and Version 3.0 of Mathematica, are provided as explained above in ``Use of the CD-ROM.''

  • Solutions notebooks. Solutions are also provided in Mathematica notebooks as explained above in ``Use of the CD-ROM.'' Students can experiment to produce new problems and graphs.

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Addresses

 The latest version of ODE.m and explanatory Mathematica notebooks are available via anonymous ftp from math.cl.uh.edu/pub/ode. The e-mail addresses of the authors are:

Alfred Gray - gray@bianchi.umd.edu
Michael Mezzino - mezzino@gauss.cl.uh.edu
Mark Pinsky - pinsky@math.nwu.edu
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Acknowledgments

 We wish to express our sincere gratitude to the following individuals who made valuable comments on various stages of the evolution of the manuscript:

  • Douglas Alexander (TRACOR, Incorporated)
  • James Anderson (University of Maryland)
  • Luis A. Cordero (University of Santiago de Compostela)
  • Albert Currier (University of Maryland)
  • Gautam Dasgupta (Columbia University)
  • Bill Davis (Ohio State University)
  • Dave Diller (Northwestern University)
  • Kit Dodson (Manchester University)
  • Chris Flannery (Northwestern University)
  • Mary Gray (American University)
  • Joe Grohens (Wolfram Research Incorporated)
  • Denny Gulick (University of Maryland)
  • Garry Helzer (University of Maryland)
  • Harry Hughes (Southern Illinois University)
  • Steve Izen (Case Western Reserve University)
  • Alfredo Jiménez (Pennsylvania State University, Hazelton)
  • Joe Kaiping (Wolfram Research Incorporated)
  • Jerry Keiper (Wolfram Research Incorporated)
  • Lee Lorch (York University)
  • William M. MacDonald (University of Maryland)
  • Kirk Mathews (Air Force Institute of Technology)
  • Ed Packel (Lake Forest College)
  • Patrick Reich (University of Houston - Clear Lake)
  • Norman Richert (University of Houston - Clear Lake)
  • Clark Robinson (Northwestern University)
  • Tarek G. Shawki (University of Illinois)
  • Renming Song (University of Michigan)
  • Nancy Stanton (Notre Dame University)
  • Volker Wihstutz (University of North Carolina)
  • Calvin Wilcox (University of Utah)
  • David Withoff (Wolfram Research Incorporated)

We also wish to express our heartfelt thanks and appreciation to our editors - Allan Wylde and Paul Wellin, for their encouragement and unwavering support during the months of development of both the text and the software. To the staff of Springer-Verlag, especially Keisha Sherbecoe and Victoria Evarretta, for their careful attention to the details of production.

And finally, to our families, who have always provided the necessary love and emotional support, especially during the challenging periods of this project.

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