This tried-and-true book of differential equations expands upon the authors' *Differential Equations: Computing and Modeling*, 2nd Edition. It covers the core concepts and techniques of elementary linear algebra—matrices and linear systems, vector spaces, eigensystems, and matrix exponentials—that are needed for a careful introduction to linear equations. Complimenting this solid foundation, the book emphasizes mathematical modeling of real-world phenomena, and offers a fresh new computational flavor evident in figures, examples, problems, and projects throughout. Chapter topics include: first order differential equations, mathematical models and numerical methods, linear systems and matrices, vector spaces, linear equations of higher order, eigenvalues and eigenvectors, linear systems of differential equations, matrix exponential methods, and nonlinear systems and phenomena. A geometric visualization for those interested in science and engineering.

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Preface

For the past half century many introductory differential equations courses for science and engineering students have emphasized the formal solution of standard types of differential equations using a (seeming) grab-bag of mechanical solution techniques. The evolution of the present text is based on experience teaching a new course with a greater emphasis on conceptual ideas and the use of computer lab projects to involve students in more intense and sustained problem-solving experiences. Both the conceptual and the computational aspects of such a course depend heavily on the perspective and techniques of linear algebra. Consequently, the study of differential equations and linear algebra in tandem reinforces the learning of both subjects. In this book we have therefore combined core topics in elementary differential equations with those concepts and methods of elementary linear algebra that are needed for a contemporary introduction to differential equations.

The availability of technical computing environments like Maple, Mathematica, and MATLAB is reshaping the current role and applications of differential equations in science and engineering, and has shaped our approach in this text. New technology motivates a shift in emphasis from traditional manual methods to both qualitative and computer-based methods that

render accessible a wider range of more realistic applications; permit the use of both numerical computation and graphical visualization to develop greater conceptual understanding; and encourage empirical investigations that involve deeper thought and analysis than standard textbook problems. Major Features

The following features of this text are intended to support a contemporary differential equations course with linear algebra that augments traditional core skills with conceptual perspectives:

The organization of the book emphasizes linear systems of algebraic and differential equations. Chapter 3 introduces matrices and determinants as needed for concrete computational purposes. Chapter 4 introduces vector spaces in preparation for understanding (in Chapter 5) the solution set of an nth order homogeneous linear differential equation as an n-dimensional vector space of functions, and for realizing that finding a general solution of the equation amounts to finding a basis for its solution space. (Students who proceed to a subsequent course in abstract linear algebra may benefit especially from this concrete prior experience with vector spaces.) Chapter 6 introduces eigenvalues and eigenvectors in preparation for solving linear systems of differential equations in Chapters 7 and 8. In Chapter 8 we may go a bit further than usual with the computation of matrix exponentials. These linear tools are applied to the analysis of nonlinear systems and phenomena in Chapter 9. We have trimmed the coverage of certain seldom-used topics and added new topics in order to place throughout a greater emphasis on core techniques as well as qualitative aspects of the subject associated with direction fields, solution curves, phase plane portraits, and dynamical systems. To this end we combine symbolic, graphic, and numeric solution methods wherever it seems advantageous. A healthy computational flavor should be evident in figures, examples, problems, and projects throughout the text. Discussions and examples of the mathematical modeling of real-world phenomena appear throughout the book. Students learn through modeling and empirical investigation to balance the questions of what equation to formulate, how to solve it, and whether a solution will yield useful information. Students also need to understand the role of existence and uniqueness theorems in the subject. While it may not be feasible to include proofs of these fundamental theorems along the way in a elementary course, students need to see precise and clear-cut statements of these theorems. We include appropriate existence and uniqueness proofs in the appendices, and occasionally refer to them in the main body of the text. Computer methods for the solution of differential equations and linear systems of equations are now common, but we continue to believe that students need to learn certain analytical methods of solution (as in Chapters 1 and 5). One reason is that effective and reliable use of numerical methods often requires preliminary analysis using standard elementary techniques; the construction of a realistic numerical model often is based on the study of a simpler analytical model. We therefore continue to stress the mastery of traditional solution techniques (especially through the inclusion of extensive problem sets). Computational Flavor

The following features highlight the computational flavor that distinguishes much of our exposition.

About 250 computer-generated graphics—over half of them new for this version of the text, and most constructed using MATLAB—show students vivid pictures of direction fields, solution curves, and phase plane portraits that bring symbolic solutions of differential equations to life. Over 30 computing projects follow key sections throughout the text. These "technology neutral" project sections illustrate the use of computer algebra systems like Maple, Mathematica, and MATLAB, and seek to actively engage students in the application of new technology. A fresh numerical emphasis that is afforded by the early introduction of numerical solution techniques in Chapter 2 (on mathematical models and numerical methods). Here and in Section 7.6, where numerical techniques for systems are treated, a concrete and tangible flavor is achieved by the inclusion of numerical algorithms presented in parallel fashion for systems ranging from graphing calculators to MATLAB. A conceptual perspective shaped by the availability of computational aids, which permits a leaner and more streamlined coverage of certain traditional manual topics (like exact equations and variation of parameters) in Chapters 1 and 5. Applications

To sample the range of applications in this text, take a look at the following questions:

What explains the commonly observed time lag between indoor and outdoor daily temperature oscillations? (Section 1.5) What makes the difference between doomsday and extinction in alligator populations? (Section 2.1) How do a unicycle and a two-axle car react differently to road bumps? (Sections 5.6 and 7.4) Why might an earthquake demolish one building and leave standing the one next door? (Section 7.4) Why might an eartquake demolish one building and leave standing the one next door? (Section 7.4) How can you predict the time of next perihelion passage of a newly observed comet? (Section 7.6) What determines whether two species will live harmoniously together, or whether competition will result in the extinction of one of them and the survival of the other? (Section 9.3) Organization and Content

The organization and content of the book may be outlined as follows:

After a precis of first-order equations in Chapter 1—with a somewhat stream-lined coverage of certain traditional symbolic methods—Chapter 2 offers an early introduction to mathematical modeling, stability and qualitative properties of differential equations, and numerical methods. This is a combination of topics that ordinarily are dispersed later in an introductory course. Chapters 3 (Linear Systems and Matrices), 4 (Vector Spaces), and 6 (Eigenvalues and Eigenvectors) provide concrete and self-contained coverage of the elementary linear algebra concepts and techniques that are needed for the solution of linear differential equations and systems. Chapter 6 concludes with applications of diagonizable matrices and a proof of the Cayley-Hamilton theorem for such matrices. Chapter 5 exploits the linear algebra of Chapters 3 and 4 to present efficiently the theory and solution of single linear differential equations. Chapter 7 is based on the eigenvalue approach to linear systems, and includes (in Section 7.5) the Jordan normal form for matrices and its application to the general Cayley-Hamilton theorem. This chapter includes an unusual number of applications (ranging from railway cars to earthquakes) of the various cases of the eigenvalue method, and concludes in Section 7.6 with numerical methods for systems. Chapter 8 is devoted to matrix exponentials with applications to linear systems of differential equations. The spectral decomposition method of Section 8.3 offers students an especially concrete approach to the computation of matrix exponentials. Our treatment of this material owes much to advice and course notes provided by Professor Dar-Veig Ho of the Georgia Institute of Technology. Chapter 9 exploits linear methods for the investigation of nonlinear systems and phenomena, and ranges from phase plane analysis to applications involving ecological and mechanical systems. Chapters 10 treats Laplace transform methods for the solution of constant-coefficient linear differential equations with a goal of handling the piecewise continuous and periodic forcing functions that are common in engineering applications. Chapter 11 treats power series methods with a goal of discussing Bessel's equation with sufficient detail for the most common elementary applications. Problems, Projects, and the Web Site

About 300 of the text's over 2000 problems are new for this book; the older problems are retained either from Edwards & Penney, Differential Equations: Computing and Modeling (Prentice Hall, 2000) or from our Elementary Linear Algebra (Prentice Hall, 1988). Each section contains a wide variety of computational exercises plus an ample number of applied or conceptual problems.

The answer section includes the answers to most odd-numbered problems and to some of the even-numbered ones. The Student Solutions Manual accompanying this book provides worked-out solutions for most of the odd-numbered problems in the book, while the Instructor's Solutions Manual provides solutions for most of the even-numbered problems as well.

The approximately 30 project sections in the text contain much additional and extended problem material designed to engage students in the exploration and application of computational technology. Most of these projects are expanded considerably in the Computing Projects Manual that accompanies the text and supplements it with additional and sometimes more challenging investigations. Each project section in this manual has parallel Using Maple, Using Mathematics, and Using MATLAB subsections that detail the applicable methods and techniques of each system, and will afford student users an opportunity to compare the merits and styles of different computational systems.

Students can download project notebooks and worksheets from the Web site prenhall/edwards/ode, where a variety of additional supporting materials ranging from reading quizzes to interactive examples and phase plane plotters are provided. Acknowledgments

In preparing this revision we profited greatly from the advice and assistance of the following very able reviewers:

Martin Forrest,

Louisiana State University

Ted Gamelin,

University of California at Los Angeles

Donald Hartig,

Calfornia Polytechnic State University at San Luis Obispo

John Henderson,

Auburn University

Gary Rosen,

Univ. of Southern California

Jeffrey Stopple,

Univ. of California at Santa Barbara

William Stout,

Salve Regina University

We thank also Bayani DeLeon for his efficient supervision of the process of book production. We owe special thanks to our editor, George Lobell, for his enthusiastic encouragement and advice, and to Dennis Kletzing for the TeX virtuosity that is evident in the attractive design and composition of this book. Once again, we are unable to express adequately our debts to Alice F. Edwards and Carol W Penney for their continued assistance, encouragement, support, and patience.

C.H.E.

hedwards@math.uga

Athens, Georgia, U.S.A.

D.E.P.

dpenney@math.uga

Athens, Georgia, U.S.A.

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