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1. The first two sections of Chapter 1 are new and include an immediate introduction to some problems that lead to differential equations and their solutions. These sections also give an early glimpse of mathematical modeling, of direction fields, and of the basic ideas of stability and instability.
2. Chapter 2 now includes a new Section 2.7 on Euler's method of numerical approximation. Another change is that most of the material on applications has been consolidated into a single section. Finally, the separate section on first order homogeneous equations has been eliminated and this material has been placed in the problem set on separable equations instead.
3. Section 4.3 on the method of undetermined coefficients for higher order equations has been simplified by using examples rather than giving a general discussion of the method.
4. The discussion of eigenvalues and eigenvectors in Section 7.3 has been shortened by removing the material relating to diagonalization of matrices and to the possible shortage of eigenvectors when an eigenvalue is repeated. This material now appears in later sections of the same chapter where the information is actually used. Sections
7.7 and 7.8 have been modified to give somewhat greater emphasis to fundamental matrices and somewhat less to problems involving repeated eigenvalues.
5. An example illustrating the instabilities that can be encountered when dealing with stiff equations has been added to Section 8.5.
6. Section 9.2 has been streamlined by considerably shortening the discussion of autonomous systems in general and including instead two examples in which trajectories can be found by integrating a single first order equation.
7. There is a new section 10.1 on two-point boundary value problems for ordinary differential equations. This material can then be called on as the method of separation of variables is developed for partial differential equations. There are also some new three-dimensional plots of solutions of the heat conduction equation and of the wave equation.
As the subject matter of differential equations continues to grow, as new technologies become commonplace, as old areas of application are expanded, and as new ones appear on the horizon, the content and viewpoint of courses and their textbooks must also evolve. This is the spirit we have sought to express in this book.
William E. Boyce Troy, New York April, 2000
It is a pleasure to offer my grateful appreciation to the many people who have generously assisted in various ways in the creation of this book.
The individuals listed below reviewed the manuscript and provided numerous valuable suggestions for its improvement:
Steven M. Baer, Arizona State University
Deborah Brandon, Carnegie Mellon University
Dante DeBlassie, Texas A & M University
Moses Glasner, Pennsylvania State University-University Park
David Gurarie, Case Western Reserve University
Don A. Jones, Arizona State University
Duk Lee, Indiana Wesleyan University
Gary M. Lieberman, Iowa State University
George Majda, Ohio State University
Rafe Mazzeo, Stanford University
Jeff Morgan, Texas A & M University
James Rovnyak, University of Virginia
L.F. Shampine, Southern Methodist University
Stan Stascinsky, Tarrant County College
Robert L. Wheeler, Virginia Tech
I am grateful to my friend of long standing, Charles Haines (Rochester Institute of Technology). In the process of revising once again the Student Solutions Manual he checked the solutions to a great many problems and was responsible for numerous corrections and improvements.
I am indebted to my colleagues and students at Rensselaer whose suggestions and reactions through the years have done much to sharpen my knowledge of differential equations as well as my ideas on how to present the subject.
My thanks also go to the editorial and production staff of John Wiley and Sons. They have always been ready to offer assistance and have displayed the highest standards of professionalism.
Most important, I thank my wife Elsa for many hours spent proofreading and checking details, for raising and discussing questions both mathematical and stylistic, and above all for her unfailing support and encouragement during the revision process. In a very real sense this book is a joint product.
William E. Boyce
Chapter 1 Introduction 1
1.1 Some Basic Mathematical Models; Direction Fields 1
1.2 Solutions of Some Differential Equations 9
1.3 Classification of Differential Equations 17
1.4 Historical Remarks 23
Chapter 2 First Order Differential Equations 29
2.1 Linear Equations with Variable Coefficients 29
2.2 Separable Equations 40
2.3 Modeling with First Order Equations 47
2.4 Differences Between Linear and Nonlinear Equations 64
2.5 Autonomous Equations and Population Dynamics 74
2.6 Exact Equations and Integrating Factors 89
2.7 Numerical Approximations: Eulerís Method 96
2.8 The Existence and Uniqueness Theorem 105