This edition, like its predecessors, is written from the viewpoint of the
applied mathematician, whose interest in differential equations may be highly
theoretical, intensely practical, or somewhere in between. We have sought to
combine a sound and accurate (but not abstract) exposition of the elementary
theory of differential equations with considerable material on methods of
solution, analysis, and approximation that have proved useful in a wide variety
of applications.
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
