现代方法与应用：微分方程导论

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"Differential Equations: An Introduction to Modern Methods and Applications by James R. Brannan and William E. Boyce" 这篇资源是关于微分方程的一本教材，由克莱姆森大学的James R. Brannan和伦斯勒理工学院的William E. Boyce合著。微分方程是数学的一个重要分支，它在物理学、工程学、生物学、经济学等多个领域都有广泛的应用。这本书旨在介绍现代微分方程的解法和应用。微分方程是描述自然界中动态系统行为的数学工具，通过建立变量之间的微分关系来研究系统的演变规律。书中的"现代方法"可能包括分离变量法、幂级数解法、特征线法、拉普拉斯变换以及非线性微分方程的解法等。"应用"部分则会涵盖实际问题的建模，如振动系统的分析、人口增长模型、热传导和流体动力学等问题。 WileyPLUS是配合该教材的在线学习平台，提供了一个创新且基于研究的学习环境，以促进有效的教学和学习。它包含互动式电子书和各种资源，旨在提高学生的学习积极性和自信心。WileyPLUS提供的特性包括： 1. **24/7技术支持**：无论何时何地，学生和教师都能获得帮助。 2. **个性化的反馈和进度证明**：增强学生的学习信心，让他们了解自己的学习进度。 3. **教师支持**：提供可靠的资源，帮助教师在课堂内外强化课程目标。 4. **预加载的作业和演示**：由主题专家创建，方便教师直接使用。 5. **2分钟教程**：快速入门指南，帮助学生和教师迅速掌握使用方法。 6. **社区资源**：教师可以与其他同行合作，寻找导师，参加虚拟或现场活动。通过WileyPLUS，学生可以体验到一个全方位、个性化的学习过程，而教师则能够更有效地管理和评估学生的学习效果。这个平台不仅提供了丰富的教学资源，还注重培养学生的自主学习能力和问题解决能力，使他们能够更好地理解和应用微分方程的知识。

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xiv Preface

of ﬁrst order equations. There are several reasons why we include introductory material on

numerical algorithms in the text.

 Euler’s method, in particular, gives great insight into the meaning of a dynamical system

(system of ﬁrst order differential equations) in that it provides a simple algorithm for

advancing the state of the dynamical system in discrete time.

 The Runge-Kutta method, while less intuitive than the Euler method, shows how a

cleverly designed algorithm can yield great rewards in terms of accuracy and efﬁciency.

 Numerical methods are an important component of the body of knowledge of differential

equations. It is conceivable that many instructors may wish to provide their students with

an introduction to some of the algorithms and their analysis.

 There are situations where a slight modiﬁcation of an elementary numerical method,

such as Euler’s method, may provide a satisfactory scheme for approximating the

solution of a problem. Illustrations of this are contained in the projects presented in

Section 2.P.4, Monte Carlo Option Pricing, and Section 4.P.4, Uniform Distribution of

Points on a Sphere.

Power Series Methods

Since numerical approximation methods are now much more efﬁcient than analytical ap-

proximations based on power series, power series methods have become less important with

the passage of time. Consequently, more and more colleges and universities are omitting the

treatment of power series methods from their ﬁrst course in differential equations. Power

series methods are important for equations having singular points, such as Bessel’s or Leg-

endre’s equations. These equations arise most naturally from solving partial differential

equations by the method of separation of variables, a topic that is usually not dealt with in

a one-semester ﬁrst course. In view of this, and consistent with the spirit of this text, we

have not included a chapter on power series methods.

Relation of This Text to Boyce and DiPrima

Brannan and Boyce is an offshoot of the well-known textbook by Boyce and DiPrima.

Readers familiar with Boyce and DiPrima will doubtless recognize in the present book

some of the hallmark features that distinguish that textbook.

To help avoid confusion among potential users of either text, the primary differences are

described below:

 Brannan and Boyce is more sharply focused on the needs of students of engineering

and science, whereas Boyce and DiPrima targets a somewhat more general audience,

including engineers and scientists.

 Brannan and Boyce is intended to be more consistent with the way contemporary scien-

tists and engineers actually use differential equations in the workplace.

 Brannan and Boyce emphasizes systems of ﬁrst order equations, introducing them earlier,

and also examining them in more detail than Boyce and DiPrima. Brannan and Boyce

has an extensive appendix on matrix algebra to support the treatment of systems in n

dimensions.

 Brannan and Boyce integrates the use of computers more thoroughly than Boyce and

DiPrima. Brannan and Boyce introduces numerical approximation methods in Chapter 1,

and assumes that most students will use computers to generate approximate solutions

and graphs throughout the book.

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Preface xv

 Brannan and Boyce emphasizes contemporary applications to agreater extent than Boyce

and DiPrima, primarily through end of chapter projects.

 Brannan and Boyce makes somewhat more use of graphs, with more emphasis on phase

plane displays, and uses engineering language (for example, state variables, transfer

functions, gain functions, and poles) to a greater extent than Boyce and DiPrima.

Options for Course Structure

Chapter dependencies are shown in the following block diagram.

Chapter 3

Systems of

Two

First Order

Equations

Chapter 2

First Order

Differential

Equations

Chapter 6

Systems of

First Order

Linear

Equations

Chapter 7

Nonlinear

Differential

Equations

and Stability

Chapter 4

Second

Order

Linear

Equations

Chapter 5

The

Laplace

Transform

Appendix A

Matrix

Algebra

Chapter 1

Introduction

The book has much built-in ﬂexibility and allows instructors to choose from many

options. Depending on the course goals of the instructor and background of the students,

selected sections may be covered lightly or even omitted.

 Chapters 5, 6, and 7 are independent of each other, and Chapters 6 and 7 are also

independent of Chapter 4. It is possible to spend much class time on one of these

chapters, or class time can be spread over two or more of them.

 The amount of time devoted to projects is entirely up to the instructor.

 For an honors class, a class consisting of students who have already had a course in

linear algebra, or a course in which linear algebra is to be emphasized, Chapter 6 may

be taken up immediately following Chapter 2. In this case, material from Appendix A,

as well as sections, examples, and problems from Chapters 3 and 4 may be selected on

an as needed, or desired, basis. This offers the possibility of spending more class time

on Chapters 5, 7 and/or selected projects.

Comments on the Use of Projects. There are several ways of using projects in the

course.

 For some of the projects, the work may be done outside of the classroom on an individual

basis.

 Some of the lengthier, more challenging projects might be more appropriately assigned

to small groups of students.

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xvi Preface

 Enough exposition is provided in a few of the projects so that students can derive

some beneﬁt from simply reading about the application. For example, students may

be interested in James Watt’s clever use of a centrifugal pendulum for controlling the

speed of a steam engine as described in Section 5.P.3, or how systems of ordinary

differential equations are used to describe wave propagation as described in Section

3.P.3. Of course, more beneﬁt may be derived by doing some or all of the project

exercises.

 Some instructors may wish to incorporate one or two projects into the course, with a

lecture devoted to each one.

 An optional strategy, say for a modeling course, is to design the course to be “project

driven”. In this approach, selected projects are used to drive the discussion of the

mathematics required to solve the problems posed in the project. The discussion of

mathematical methods and theory are then intertwined with the project. This might be

appropriate for some of the projects that are highly integrative in nature. For exam-

ple, virtually every topic in Chapter 6, and then some, is utilized to solve the problem

posed in Section 6.P.3, Controlling a Spring-Mass System to Equilibrium. Similarly,

most of the machinery of Chapter 5 is required to solve the problem in Section 5.P.3,

The Watt Governor, Feedback Control, and Stability. The projects then serve to motivate

the mathematics.

 Projects can be given as “extra credit” assignments.

Acknowledgments

It is a pleasure to offer our grateful appreciation to the many people who have generously

assisted in the preparation of this book.

To the individuals listed below who reviewed parts or all of the manuscript at various

stages of its development:

Adam Boucher, University of New Hampshire

Benito Chen, University of Wyoming

David Dwiggns, University of Memphis

Marek Elzanowski, Portland State University

William Emerson, Metro State College, Denver

Joerg Feldvoss, University of Southern Alabama

David Gurarie, Case Western Reserve University

Thomas Hill, Lafayette College

Matthias Kawski, Arizona State University

Todd King, Michigan Technological University

Richard Le Borne, Tennessee Tech University

Sean Lynch, University of Illinois Chicago

Tom Morley, Georgia Tech

William F. Moss, Clemson University

Maury Rahe, Texas A&M University

Leela Rakesh, Central Michigan University

Eric De Sturler, Virginia Tech

Blake Thornton, Washington University, St. Louis

Arthur G. Wasserman, University of Michigan

Shangyou Zhang, University of Delaware

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xviii Preface

Supplemental Resources for Instructors and Students

An Instructor’s Solutions Manual, ISBN 978-0-470-45826-6 includes solutions for all

problems in the text.

A Student Solutions Manual, ISBN 978-0-470-45825-9 includes solutions for selected

problems in the text.

A Companion Web site, www.wiley.com/college/brannan, provides a wealth of resources

for students and instructors, including:

 PowerPoint slides of important ideas and graphics for study and note taking.

 Review and Study Outlines to help students prepare for quizzes and exams.

 Online Review Quizzes to enable students to test their knowledge of key concepts. For

further review diagnostic feedback is provided that refers to pertinent sections in the text.

 Additional problems for use with Mathematica, Maple, and Mat la b , allowing oppor-

tunities for further exploration of important ideas in the course utilizing these computer

algebra and numerical analysis packages.

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