This book presents analytical, graphical and numerical methods in a unified way—as methods of solution and as means of illuminating concepts. Numerical methods are introduced in the first chapter, interpreted in the light of graphics, and provide the core theme around which the first seven chapters revolve. These chapter titles are: The First Order Equation *y = f(x,y); First Order Systems Introduction; Higher Order Linear Equations; First Order Systems—Linear Methods; Series Methods and Famous Functions; and Bifurcations and Chaos. The other three chapters cover the laplace transform; partial differential equations and fourier series; and the finite differences method. A unique combination of the traditional topics of differential equations and computer graphics, for anyone interested in taking advantage of this learning package.*

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Preface

Some time ago I searched for a textbook for a sophomore course in differential equations that would combine analytical (algebraic) methods of solution with graphical and numerical methods in a unified way. Some texts made computer graphics the center of the course and left out such topics as variation of parameters and infinite series. Other texts retained the traditional topics, but the graphics seemed to be grafted on as an afterthought. This book is an outgrowth of this failed search. The book retains almost all the traditional canon of differential equations, but it employs graphical and numerical methods from the outset, both as methods of solution and as means of illuminating concepts.

To employ graphical and numerical methods from the start, it was necessary to make first-order systems and reduction to first-order systems the focal point. First-order systems form the core subject matter of Chapters 1 through 5 and Chapter 7. Chapter 6 covers power series solutions, but even here first-order systems make a brief appearance in order to make clear why points at which the leading coefficient of a linear differential equation vanishes must be considered singular. Through first-order systems, solutions can easily be presented graphically with today's computer resources. This opens the way for visual interpretation of solutions and fields. First-order systems also provide the unified means of applying numerical methods to a very wide range of differential equations. Because of this, differential equations can be investigated that could not be considered in times gone by. Models of competing species, the pendulum, and the tunnel diode oscillator are taken up early in the text.

In spite of the emphasis on first-order systems, I have not neglected the basics of analytic solutions. Separable, linear, and exact equations are solved in the study of a single first-order equation in Chapter 2, and higher-order constant coefficient linear equations are treated in Chapter 4. However, the knowledge of first-order systems developed in Chapter 3 is used to establish the strategy for solving higher-order linear equations. Power series methods are also not neglected. Indeed, they cannot be, since they are needed in the solution of partial differential equations, which is the subject of Chapter 9.

Chapter 9 presents the solution of partial differential equations through the method of separation of variables and Fourier series. Chapter 10 introduces the reader to numerical methods of solution for partial differential equations. These two chapters were more difficult to write than the others because there is no unifying theme, such as first-order systems for ordinary differential equations. Nonetheless, graphics and numerical methods have been employed to help clarify ideas and to extend the range of equations solved. Computer algebra systems (CAS) such as Mathematica, Maple or MatLab (the Three M's) are used to advantage to illustrate convergence of Fourier series, graph modes of vibration for drumheads, and animate solutions. The chapter on numerical methods for partial differential equations is, I think, new in a book of this type. However, I believe it is entirely in keeping with the theme of this book and the availability of powerful computing resources. The use of a CAS makes the instability of some of the finite difference methods easy to explore, and it makes possible the exploration of some nonlinear partial differential equations.

Chapter 8 is a traditional treatment of the Laplace transform. The Laplace transform does not call for graphical or numerical methods, but I thought it important to include the Laplace transform because it is such an elegant way of dealing with constant coefficient linear equations and discontinuous forcing functions.

A large proportion of the exercises call for the use of a computer. The necessary software is available at the Prentice Hall web site: prenhall/banks A Note to Instructors

For ordinary differential equations I have written a series of applications specific to the task at hand for the Windows and MacOs operating systems. These require no other support than the operating system. Packages for the entire book are also available at the web site for each of the Three M's. These routines are also listed completely in the instructors manual. If, however, you wish to have your students develop their own packages in one of the Three M's, Appendix D provides a guide to the development of these packages. This does have pedagogical value if you have the time. If you intend to cover the material on partial differential equations, then developing facility with one of the Three M's will be helpful.

The formal abstractions of vector spaces and linear transformations are introduced only in Section 9.4. Admittedly, this is quite late in the text, but it is not until this point that the references to linear combinations and linearity properties have sufficiently motivated the abstractions of a vector space and linear transformation. Together, Appendix A and Section 9.4 provide an introduction to the basic elements of linear algebra. They are self-contained, and Appendix A has exercises also. Thus Appendix A and Section 9.4 can be used at any time as supplementary lectures. If you feel the need to introduce concepts from linear algebra earlier, there is nothing to prevent you doing so. Indeed, if your students lack a good college algebra preparation, I recommend covering the material on solutions of systems of linear equations in Appendix A before going into Chapter 4.

The Supplementary Exercise sections at the end of chapters contain exercises that are of a more challenging nature or develop a topic that was not covered in the chapter. These exercises could be used as projects to be completed in, say, a ten-day period.

I confess to repeatedly abusing notation by referring to f(x) as a function as well as the value of the function. I do this deliberately because I believe that using the precise notation for functions would tend to confuse students who are not yet mathematically very sophisticated. Besides, the precise notation can become quite cumbersome at times. Students will find "consider the function f(x) = 3x..." much more palatable than "consider the function f : R ® R defined by f(x) = 3x..."

It is intended that the book be covered in two semesters. The first semester should aim to cover Chapters 1 through 7 and the second Chapters 8 through 10. In one quarter, I have been able to cover Chapters 1 through 4 and Sections 6.1 and 6.2 with the omission of a few nonessential sections. There is no question that the material on partial differential equations is inherently more difficult than the rest of the text. However, this material is accessible if it is covered at a gentle pace. Sections that may be omitted without loss of continuity are 2.5, 3.5, 3.6, and all of Chapters 7, 8, and 10.

Finally, detailed suggestions for teaching sections of the text may be found in the instructors manual for the text. Also included are suggestions for laboratory activities, sample exam questions, and complete listings of the software packages. Acknowledgments

Several of my colleagues have taught from the text during its preparation. Their contributions have been invaluable. I would like to thank Dr. Weiqing Xie, Dr. Martin Nakashima, Dr. Alan Radnitz, and especially Dr. Harriet Lord for her constant help in the development of this text. I would also like to thank my editor, George Lobell, and his staff for their guidance and help.

Bernard W. Banks

bwbanks@csupomona

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