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# Elementary Differential Equations and Boundary Value Problems / Edition 8

Elementary Differential Equations and Boundary Value Problems / Edition 8 available in Paperback

- ISBN-10:
- 0471644544
- ISBN-13:
- 9780471644545
- Pub. Date:
- 04/16/2004
- Publisher:
- Wiley, John & Sons, Incorporated

## Paperback

###### ADVERTISEMENT

## Overview

Details the methods for solving ordinary and partial differential equations. New material on limit cycles, the Lorenz equations and chaos has been added along with nearly 300 new problems. Also features expanded discussions of competing species and predator-prey problems plus extended treatment of phase plane analysis, qualitative methods and stability.

## Product Details

ISBN-13: | 9780471644545 |
---|---|

Publisher: | Wiley, John & Sons, Incorporated |

Publication date: | 04/16/2004 |

Pages: | 784 |

## About the Author

**William E. Boyce**received his B.A. degree in Mathematics from Rhodes College, and his M.S. and Ph.D. degrees in Mathematics from Carnegie-Mellon University. He is a member of the American Mathematical Society, the Mathematical Association of America, and the Society of Industrial and Applied Mathematics. He is currently the Edward P. Hamilton Distinguished Professor Emeritus of Science Education (Department of Mathematical Sciences) at Rensselaer. He is the author of numerous technical papers in boundary value problems and random differential equations and their applications. He is the author of several textbooks including two differential equations texts, and is the coauthor (with M.H. Holmes, J.G. Ecker, and W.L. Siegmann) of a text on using Maple to explore Calculus. He is also coauthor (with R.L. Borrelli and C.S. Coleman) of

*Differential Equations Laboratory Workbook*(Wiley 1992), which received the EDUCOM Best Mathematics Curricular Innovation Award in 1993. Professor Boyce was a member of the NSF-sponsored CODEE (Consortium for Ordinary Differential Equations Experiments) that led to the widely-acclaimed ODE Architect. He has also been active in curriculum innovation and reform. Among other things, he was the initiator of the "Computers in Calculus" project at Rensselaer, partially supported by the NSF. In 1991 he received the William H. Wiley Distinguished Faculty Award given by Rensselaer.

**Richard C. DiPrima** (deceased) received his B.S., M.S., and Ph.D.
degrees in Mathematics from Carnegie-Mellon University. He joined the faculty of
Rensselaer Polytechnic Institute after holding research positions at MIT,
Harvard, and Hughes Aircraft. He held the Eliza Ricketts Foundation
Professorship of Mathematics at Rensselaer, was a fellow of the American Society
of Mechanical Engineers, the American Academy of Mechanics, and the American
Physical Society. He was also a member of the American Mathematical Society, the
Mathematical Association of America, and the Society of Industrial and Applied
Mathematics. He served as the Chairman of the Department of Mathematical
Sciences at Rensselaer, as President of the Society of Industrial and Applied
Mathematics, and as Chairman of the Executive Committee of the Applied Mechanics
Division of ASME. In 1980, he was the recipient of the William H. Wiley
Distinguished Faculty Award given by Rensselaer. He received Fulbright
fellowships in 1964-65 and 1983 and a Guggenheim fellowship in 1982-83. He was
the author of numerous technical papers in hydrodynamic stability and
lubrication theory and two texts on differential equations and boundary value
problems. Professor DiPrima died on September 10, 1984.

## Table of Contents

Preface | vii | |

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: Euler's Method | 96 |

2.8 | The Existence and Uniqueness Theorem | 105 |

2.9 | First Order Difference Equations | 115 |

Chapter 3 | Second Order Linear Equations | 129 |

3.1 | Homogeneous Equations with Constant Coefficients | 129 |

3.2 | Fundamental Solutions of Linear Homogeneous Equations | 137 |

3.3 | Linear Independence and the Wronskian | 147 |

3.4 | Complex Roots of the Characteristic Equation | 153 |

3.5 | Repeated Roots; Reduction of Order | 160 |

3.6 | Nonhomogeneous Equations; Method of Undetermined Coefficients | 169 |

3.7 | Variation of Parameters | 179 |

3.8 | Mechanical and Electrical Vibrations | 186 |

3.9 | Forced Vibrations | 200 |

Chapter 4 | Higher Order Linear Equations | 209 |

4.1 | General Theory of nth Order Linear Equations | 209 |

4.2 | Homogeneous Equations with Constant Coeffients | 214 |

4.3 | The Method of Undetermined Coefficients | 222 |

4.4 | The Method of Variation of Parameters | 226 |

Chapter 5 | Series Solutions of Second Order Linear Equations | 231 |

5.1 | Review of Power Series | 231 |

5.2 | Series Solutions near an Ordinary Point, Part I | 238 |

5.3 | Series Solutions near an Ordinary Point, Part II | 249 |

5.4 | Regular Singular Points | 255 |

5.5 | Euler Equations | 260 |

5.6 | Series Solutions near a Regular Singular Point, Part I | 267 |

5.7 | Series Solutions near a Regular Singular Point, Part II | 272 |

5.8 | Bessel's Equation | 280 |

Chapter 6 | The Laplace Transform | 293 |

6.1 | Definition of the Laplace Transform | 293 |

6.2 | Solution of Initial Value Problems | 299 |

6.3 | Step Functions | 310 |

6.4 | Differential Equations with Discontinuous Forcing Functions | 317 |

6.5 | Impulse Functions | 324 |

6.6 | The Convolution Integral | 330 |

Chapter 7 | Systems of First Order Linear Equations | 339 |

7.1 | Introduction | 339 |

7.2 | Review of Matrices | 348 |

7.3 | Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors | 357 |

7.4 | Basic Theory of Systems of First Order Linear Equations | 368 |

7.5 | Homogeneous Linear Systems with Constant Coefficients | 373 |

7.6 | Complex Eigenvalues | 384 |

7.7 | Fundamental Matrices | 393 |

7.8 | Repeated Eigenvalues | 401 |

7.9 | Nonhomogeneous Linear Systems | 411 |

Chapter 8 | Numerical Methods | 419 |

8.1 | The Euler or Tangent Line Method | 419 |

8.2 | Improvements on the Euler Method | 430 |

8.3 | The Runge-Kutta Method | 435 |

8.4 | Multistep Methods | 439 |

8.5 | More on Errors; Stability | 445 |

8.6 | Systems of First Order Equations | 455 |

Chapter 9 | Nonlinear Differential Equations and Stability | 459 |

9.1 | The Phase Plane; Linear Systems | 459 |

9.2 | Autonomous Systems and Stability | 471 |

9.3 | Almost Linear Systems | 479 |

9.4 | Competing Species | 491 |

9.5 | Predator-Prey Equations | 503 |

9.6 | Liapunov's Second Method | 511 |

9.7 | Periodic Solutions and Limit Cycles | 521 |

9.8 | Chaos and Strange Attractors; the Lorenz Equations | 532 |

Chapter 10 | Partial Differential Equations and Fourier Series | 541 |

10.1 | Two-Point Boundary Valve Problems | 541 |

10.2 | Fourier Series | 547 |

10.3 | The Fourier Convergence Theorem | 558 |

10.4 | Even and Odd Functions | 564 |

10.5 | Separation of Variables; Heat Conduction in a Rod | 573 |

10.6 | Other Heat Conduction Problems | 581 |

10.7 | The Wave Equation; Vibrations of an Elastic String | 591 |

10.8 | Laplace's Equation | 604 |

Appendix A. | Derivation of the Heat Conduction Equation | 614 |

Appendix B. | Derivation of the Wave Equation | 617 |

Chapter 11 | Boundary Value Problems and Sturm-Liouville Theory | 621 |

11.1 | The Occurrence of Two Point Boundary Value Problems | 621 |

11.2 | Sturm-Liouville Boundary Value Problems | 629 |

11.3 | Nonhomogeneous Boundary Value Problems | 641 |

11.4 | Singular Sturm-Liouville Problems | 656 |

11.5 | Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion | 663 |

11.6 | Series of Orthogonal Functions: Mean Convergence | 669 |

Answers to Problems | 679 | |

Index | 737 |

## Preface

The book is written primarily for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for reading the book is a working knowledge of calculus, gained from a normal two- or three-semester course sequence or its equivalent.

**Mathematical Modeling**

The main reason for solving many differential equations is to try to learn something about an underlying physical process that the equation is believed to model. It is basic to the importance of differential equations that even the simplest equations correspond to useful physical models, such as exponential growth and decay, spring-mass systems, or electrical circuits. Gaining an understanding of a complex natural process is usually accomplished by combining or building upon simpler and more basic models. Thus a thorough knowledge of these models, the equations that describe them, and their solutions, is the first and indispensable step toward the solution of more complex and realistic problems.

More difficult problems often require the use of a variety of tools, both analytical and numerical. We believe strongly that pencil and paper methods must be combined with effective use of a computer. Quantitative results and graphs, often produced by a computer, serve to illustrate and clarify conclusions that may be obscured by complicated analytical expressions. On the other hand, the implementation of an efficient numerical procedure typically rests on a good deal of preliminary analysis - to determine the qualitative features of the solution as a guide to computation, to investigate limiting or special cases, or to discover which ranges of the variables or parameters may require or merit special attention.

Thus, a student should come to realize that investigating a difficult problem may well require both analysis and computation; that good judgment may be required to determine which tool is best-suited for a particular task; and that results can often be presented in a variety of forms.

**A Flexible Approach**

To be widely useful a textbook must be adaptable to a variety of instructional strategies. This implies at least two things. First, instructors should have maximum flexibility to choose both the particular topics that they wish to cover and also the order in which they want to cover them. Second, the book should be useful to students having access to a wide range of technological capability.

With respect to content, we provide this flexibility by making sure that, so far as possible, individual chapters are independent of each other. Thus, after the basic parts of the first three chapters are completed (roughly Sections 1.1 through 1.3, 2.1 through 2.5, and 3.1 through 3.6) the selection of additional topics, and the order and depth in which they are covered, is at the discretion of the instructor. For example, while there is a good deal of material on applications of various kinds, especially in Chapters 2, 3, 9, and 10, most of this material appears in separate sections, so that an instructor can easily choose which applications to include and which to omit. Alternatively, an instructor who wishes to emphasize a systems approach to differential equations can take up Chapter 7 (Linear Systems) and perhaps even Chapter 9 (Nonlinear Autonomous Systems) immediately after Chapter 2. Or, while we present the basic theory of linear equations first in the context of a single second order equation (Chapter 3), many instructors have combined this material with the corresponding treatment of higher order equations (Chapter 4) or of linear systems (Chapter 7). Many other choices and combinations are also possible and have been used effectively with earlier editions of this book.

With respect to technology, we note repeatedly in the text that computers are extremely useful for investigating differential equations and their solutions, and many of the problems are best approached with computational assistance. Nevertheless, the book is adaptable to courses having various levels of computer involvement, ranging from little or none to intensive. The text is independent of any particular hardware platform or software package. More than 450 problems are marked with the symbol 1 to indicate that we consider them to be technologically intensive. These problems may call for a plot, or for substantial numerical computation, or for extensive symbolic manipulation, or for some combination of these requirements. Naturally, the designation of a problem as technologically intensive is a somewhat subjective judgment, and the 1 is intended only as a guide. Many of the marked problems can be solved, at least in part, without computational help, and a computer can be used effectively on many of the unmarked problems.

From a student's point of view, the problems that are assigned as homework and that appear on examinations drive the course. We believe that the most outstanding feature of this book is the number, and above all the variety and range, of the problems that it contains. Many problems are entirely straightforward, but many others are more challenging, and some are fairly open-ended, and can serve as the basis for independent student projects. There are far more problems than any instructor can use in any given course, and this provides instructors with a multitude of possible choices in tailoring their course to meet their own goals and the needs of their students.

One of the choices that an instructor now has to make concerns the role of computing in the course. For instance, many more or less routine problems, such as those requesting the solution of a first or second order initial value problem, are now easy to solve by a computer algebra system. This edition includes quite a few such problems, just as its predecessors did. We do not state in these problems how they should be solved, because we believe that it is up to each instructor to specify whether their students should solve such problems by hand, with computer assistance, or perhaps both ways. Also, there are many problems that call for a graph of the solution. Instructors have the option of specifying whether they want an accurate computer-generated plot or a hand-drawn sketch, or perhaps both.

There are also a great many problems, as well as some examples in the text, that call for conclusions to be drawn about the solution. Sometimes this takes the form of asking for the value of the independent variable at which the solution has a certain property. Other problems ask for the effect of variations in a parameter, or for the determination of a critical value of a parameter at which the solution experiences a substantial change. Such problems are typical of those that arise in the applications of differential equations, and, depending on the goals of the course, an instructor has the option of assigning few or many of these problems.

**Supplementary Materials**

Three software packages that are widely used in differential equations courses are Maple, Mathematica, and Matlab. The books Differential Equations with Maple, Differential Equations with Mathematica, and Differential Equations with Matlab by K. R. Coombes, B. R. Hunt, R. L. Lipsman, J. E. Osborn, and G. J. Stuck, all at the University of Maryland, provide detailed instructions and examples on the use of these software packages for the investigation and analysis of standard topics in the course.

For the first time, this text is available in an Interactive Edition, featuring an eBook version of the text linked to the award-winning ODE Architect. The interactive eBook links live elements in each chapter to ODE Architect's powerful, yet easy-to-use, numerical differential equations solver and multimedia modules. The eBook provides a highly interactive environment in which students can construct and explore mathematical models using differential equations to investigate both real-world and hypothetical situations. A companion e-workbook that contains additional problems sets, called Explorations, provides background and opportunities for students to extend the ideas contained in each module. A stand-alone version of ODE Architect is also available.

There is a Student Solutions Manual, by Charles W. Haines of Rochester Institute of Technology, that contains detailed solutions to many of the problems in the book. A complete set of solutions, prepared by Josef Torok of Rochester Institute of Technology, is available to instructors via the Wiley website at www.wiley.com/college/Boyce.

**Important Changes in the Seventh Edition**

Readers who are familiar with the preceding edition will notice a number of modifications, although the general structure remains much the same. The revisions are designed to make the book more readable by students and more usable in a modern basic course on differential equations. Some changes have to do with content; for example, mathematical modeling, the ideas of stability and instability, and numerical approximations via Euler's method appear much earlier now than in previous editions. Other modifications are primarily organizational in nature. Most of the changes include new examples to illustrate the underlying ideas.

1. The first two sections of Chapter I 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