利用Mathematica探索微分方程学习指南

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《基于MATHEMATICA的微分方程学习》第三版是一本专为利用Mathematica这一强大的计算工具学习微分方程而设计的教材。本书由Martha L. Abell和James P. Braselton合著，旨在通过Mathematica的实际操作，帮助读者深入理解并掌握微分方程的基础理论与应用技巧。在本书中，作者详细介绍了如何利用Mathematica的强大可视化功能来解决和分析各种类型的微分方程，包括线性方程、常微分方程、偏微分方程，以及更复杂的动态系统。读者不仅能学习到微分方程的基本概念，如阶数、解的概念和分类，还能体验到如何运用Mathematica进行符号计算、数值模拟、图形绘制和数据可视化。章节内容可能包括但不限于以下几点： 1. **Mathematica入门**：介绍Mathematica的基本界面、编程语法和环境设置，为后续的微分方程处理打下基础。 2. **微分方程基础知识**：这部分涵盖微分方程的基本定义、解的存在性和唯一性理论，以及初等微分方程的解法。 3. **线性微分方程**：讲解如何使用Mathematica求解一阶和高阶线性微分方程，包括特征根和特征向量的方法。 4. **非线性微分方程**：探讨如何处理如牛顿-莱布尼茨公式、欧拉方法和龙格-库塔方法等非线性问题，并通过Mathematica实例展示求解过程。 5. **偏微分方程与傅里叶变换**：介绍二维和三维偏微分方程的表示与求解，以及Mathematica中的PDE工具包。 6. **动态系统与混沌理论**：通过Mathematica模拟洛伦兹吸引子等复杂系统的动力学行为，理解混沌现象。 7. **数值方法与误差分析**：讨论有限差分法、辛普森法则等数值求解方法，并演示如何评估计算误差。 8. **实际应用示例**：书中会提供多个现实生活中的应用案例，如物理学、工程学、经济学等领域中的微分方程模型，以加深理论理解。此外，书中还可能包含关于如何使用Mathematica的交互式教程、编程技巧、以及如何利用其强大的图形功能来清晰地展示解的空间结构和时间演化。《基于MATHEMATICA的微分方程学习》是一本既实用又深入的教程，旨在通过Mathematica这一平台，使读者在微分方程的学习过程中享受到直观、高效的体验。

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Preface xv

also be carried out by Mathematica in a timely manner on a computer as powerful

as a Macintosh IIcx.

Now, we are on top of the world with Power Macintosh G4’s with 768 megs

of RAM and 50 gig hard drives, which will almost certainly be obsolete by the

time you are reading this. The examples presented in Differential Equations with

Mathematica continue to be the ones that we think are most similar to the prob-

lems encountered by beginning users and are presented in the context of someone

familiar with mathematics typically encountered by undergraduates. However,

for this third edition of Differential Equations with Mathematica we have taken the

opportunity to expand on several of our favorite examples because the machines

now have the speed and power to explore them in greater detail.

Other improvements to the third edition include:

1. Throughout the text, we have attempted to eliminate redundant examples

and added several interesting ones. The following changes are especially

worth noting.

(a) In Chapter 2, First-Order Ordinary Differential Equations, we present

the integrating factor approach, variation of parameters, and method

of undetermined coefﬁcients when solving ﬁrst-order linear equations.

(b) In Chapter 3, we discuss the Logistic difference equation and give

some surprisingly simple ways to generate the classic “Pitchfork dia-

gram” with Mathematica.

a new section on nonlinear equations has been added.

(d) Chapter 5, Applications of Higher-Order Equations, has also been

completely reorganized. The catenary is now included in the Other

Applications section.

(e) Chapter 6, Systems of Ordinary Differential Equations, includes sev-

eral new examples. See especially Example 6.2.5.

(f) Chapter 7, Applications of Systems, includes several new examples.

See especially Examples 7.3.3, 7.3.4, and 7.3.6.

(g) We have included references that we ﬁnd particularly interesting in

the Bibliography, even if they are not speciﬁc Mathematica-related

texts. A comprehensive list of Mathematica-related publications can

be found at the Wolfram website.

http://store.wolfram.com/catalog/books/.

Finally, we must express our appreciation to those who assisted in this project.

We would like to express appreciation to our editors, Tom Singer and Barbara

Holland, and our production editor, Brandy Palacios, at Academic Press for pro-

viding a pleasant environment in which to work. In addition, Wolfram Research,

2 Chapter 1 Introduction to Differential Equations

modeled with differential equations. For example, we will see that the motion of a

pendulum can be modeled by a differential equation. When we solve the problem

of the motion of a pendulum, we use technology to actually watch the pendulum

move. The same is true for the motion of a mass attached to the end of a spring

as well as many other problems. In having this ability, the study of differential

equations becomes much more meaningful as well as interesting.

If you are a beginning Mathematica user and, especially, new to Version 5.0, the

Numerous references like

Abell and Braselton’s

Mathematica By Example [1]

are also available to beginning

users of Mathematica.

Appendix contains an introduction to Mathematica, including discussions about

entering and evaluating commands, loading packages, and taking advantage of

Mathematica’s extensive help facility.

Although Chapter 1 is short in length, Chapter 1 introduces examples that will

be investigated in subsequent chapters. Also, the vocabulary introduced in Chap-

ter 1 will be used throughout the text. Consequently, even though, to a large extent,

it may be read quickly, subsequent chapters will take advantage of the terminol-

ogy and techniques discussed here.

1.1 Deﬁnitions and Concepts

We begin our study of differential equations by explaining what a differential

equation is.

Deﬁnition 1 (Differential Equation). A differential equation is an equation that

contains the derivative or differentials of one or more dependent variables with respect to

one or more independent variables. If the equation contains only ordinary derivatives (of

one or more dependent variables) with respect to a single independent variable, the equation

is called an ordinary differential equation.

EXAMPLE 1.1.1: Thus, dy/dx  x



cos y



and dy/ dx  du/dx  u  x

are examples of ordinary differential equations.

The equation y1dx  x cos ydy  1 is an ordinary differential equation

written in differential form.

Using prime notation,asolution of the ordinary differential equation xy











 n



y  0, which is called Bessel’s equation, is a function y 

yx with the property that xd

y/dx

 xdy/dx



 n



y is identically

the 0 function.

1.1 Deﬁnitions and Concepts 3

On the other hand,











a  byx

m  nxy

(1.1)

where a, b, m,andn are positive constants, is a system of two ordinary

differential equations, called the predator–prey equations.Asolution

See texts like Giordano,

Weir, and Fox’s A First Course

in Mathematical Modeling [12]

and similar texts for detailed

descriptions of

predator–prey models.

consists of two functions x  xt and y  yt that satisfy both equations.

Predator–prey models can exhibit very interesting behavior as we will

see when we study systems in more detail.

Note that a system of differential equations can consist of more than

two equations. For example, the basic equations that describe the com-

petition between two organisms, with population densities x

and x

respectively, in a chemostat are

See Smith and Waltman’s The

Theory of the Chemostat [24]

for a detailed discussion of

chemostat models.













 1  S 

 S



 S



 x



 S

 1





 x



 S

 1



(1.2)

where



denotes differentiation with respect to t; S  St, x

 x

t,

and x

 x

t. For equations (1.2), we remark that S denotes the con-

centration of the nutrient available to the competitors with population

densities x

and x

. We investigate chemostat models in more detail in

Chapter 9.

If the equation contains partial derivatives of one or more dependent variables,

then the equation is called a partial differential equation.

EXAMPLE 1.1.2: Because the equations involve partial derivatives of

an unknown function, equations like u

u

t



u

x

and uu

 u  u

are

partial differential equations. For Laplace’s equation,



x





y

 0 a

solution would be a function u  ux, y such that u

 u

is identically

the 0 function. A solution u  ux, t of the wave equation is a function

satisfying



t





x

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