1、1Mathematical EnglishNew Words&Expressions:approximate evaluation 近似估计近似估计 initial 初始的初始的disintegrate 解体,衰变解体,衰变 integrate 求积分求积分 differentiable 可微的可微的 polynomial 多项式多项式exponential 指数的指数的 rational function 有理函数有理函数2.9 微分方程简介微分方程简介Introduction to Differential Equations2Mathematical EnglishNew Words&E
2、xpressions:differential equation 微分方程微分方程partial differential equation 偏微分方程偏微分方程inverse trigonometric function 反三角函数反三角函数approximate evaluation 近似估计近似估计initial value problem 初值问题初值问题initial condition 初始条件初始条件mathematical physics 数学物理数学物理boundary condition 边界条件边界条件2.9 Introduction to Differential Eq
3、uations3Mathematical EnglishKey points:4Mathematical English 1.理解微分方程的分类。理解微分方程的分类。2.理解矩阵学习的重要性。理解矩阵学习的重要性。5Mathematical EnglishA large variety of scientific problems arise in which one tries to determine something from its rate of change.9-A Introduction大量的科学问题需要人们根据事物的变化率来确定该大量的科学问题需要人们根据事物的变化率来确定
4、该事物。事物。For example,we could try to compute the position of a moving particle from a knowledge of its velocity or acceleration.例如,我们可以由已知速度或者加速度来计算移动质例如,我们可以由已知速度或者加速度来计算移动质点的位置点的位置.6Mathematical EnglishOr a radioactive substance may be disintegrating at a known rate and we may be required to determi
5、ne the amount of material present after a given time.又如,某种放射性物质可能正在以已知的速度进行衰又如,某种放射性物质可能正在以已知的速度进行衰变,需要我们确定在给定的时间后遗留物质的总量。变,需要我们确定在给定的时间后遗留物质的总量。7Mathematical EnglishIn examples like these,we are trying to determine an unknown function from prescribed information expressed in the form of an equation
6、 involving at least one of the derivatives of the unknown function.在类似的例子中,我们力求由方程的形式表述的信息在类似的例子中,我们力求由方程的形式表述的信息来确定未知函数,而这种方程至少包含了未知函数的来确定未知函数,而这种方程至少包含了未知函数的一个导数。一个导数。8Mathematical EnglishThese equations are called differential equations,and their study forms one of the most challenging branches o
7、f mathematics.这些方程称为微分方程,对其研究形成了数学中最具这些方程称为微分方程,对其研究形成了数学中最具有挑战性的一个分支。有挑战性的一个分支。Differential equations are classified under two main headings:ordinary and partial,depending on whether the unknown is a function of just one variable or of two or more variables.微分方程根据未知量是单变量函数还是多变量函数分微分方程根据未知量是单变量函数还是多
8、变量函数分成两个主题:常微分方程和偏微分方程。成两个主题:常微分方程和偏微分方程。9Mathematical EnglishA simple example of an ordinary differential equation is the relation f(x)=f(x)(9.1)which is satisfied,in particular by the exponential function,f(x)=ex.常微分方程的一个简单例子是常微分方程的一个简单例子是 f(x)=f(x),特别地,特别地,指数函数指数函数f(x)=ex 满足这个等式。满足这个等式。We shall s
9、ee presently that every solution of(9.1)must be of the form f(x)=C ex,where C may be any constant.我们马上就会发现我们马上就会发现(9.1)的每一个解都一定是的每一个解都一定是f(x)=Cex这种形式,这里这种形式,这里C可以是任何常数。可以是任何常数。10Mathematical EnglishOn the other hand,an equation likeis an example of a partial differential equation.另一方面,如下方程是偏微分方程的一个例
10、子。另一方面,如下方程是偏微分方程的一个例子。This particular one,is called Laplaces equation,appears in the theory of electricity and magnetism,fluid mechanics,and elsewhere.这个特殊的方程叫做拉普拉斯方程,出现于电磁学理这个特殊的方程叫做拉普拉斯方程,出现于电磁学理论、流体力学理论以及其他理论中。论、流体力学理论以及其他理论中。2222(,)(,)0f x yf x yxy11Mathematical EnglishThe study of differential
11、 equations is one part of mathematics that,perhaps more than any other,has been directly inspired by mechanics,astronomy,and mathematical physics.微分方程的研究是数学的一部分,也许比其他分支更微分方程的研究是数学的一部分,也许比其他分支更多的直接受到力学,天文学和数学物理的推动。多的直接受到力学,天文学和数学物理的推动。Its history began in the 17th century when Newton,Leibniz,and the
12、Bernoullis solved some simple differential equations arising from problems in geometry and mechanics.微分方程起源于微分方程起源于17世纪,当时牛顿,莱布尼茨,伯努世纪,当时牛顿,莱布尼茨,伯努利家族解决了一些来自几何和力学的简单的微分方程。利家族解决了一些来自几何和力学的简单的微分方程。12Mathematical EnglishThese early discoveries,beginning about 1690,gradually led to the development of a
13、lot of“special tricks”for solving certain special kinds of differential equation.开始于开始于1690年的早期发现,逐渐导致了解某些特殊类年的早期发现,逐渐导致了解某些特殊类型的微分方程的大量特殊技巧的发展。型的微分方程的大量特殊技巧的发展。13Mathematical EnglishAlthough these special tricks are applicable in relatively few cases,they do enable us to solve many differential equ
14、ations that arise in mechanics and geometry,so their study is of practical importance.尽管这些特殊的技巧只是适用于相对较少的几种情况,尽管这些特殊的技巧只是适用于相对较少的几种情况,但他们能够解决许多出现于力学和几何中的微分方程,但他们能够解决许多出现于力学和几何中的微分方程,因此,他们的研究具有重要的实际应用。因此,他们的研究具有重要的实际应用。14Mathematical EnglishSome of these special methods and some of the problems which
15、 they help us solve are discussed near the end of this chapter.这些特殊的技巧和利用这些技巧可以解决的一些问题这些特殊的技巧和利用这些技巧可以解决的一些问题将在本章最后讨论。将在本章最后讨论。15Mathematical EnglishExperience has shown that it is difficult to obtain mathematical theories of much generality about solution of differential equations,except for a few
16、types.经验表明除了几个典型方程外,很难得到微分方程解经验表明除了几个典型方程外,很难得到微分方程解的一般性数学理论。的一般性数学理论。16Mathematical EnglishThe simplest types of linear differential equations and some of their applications are also discussed in this introductory chapter.A more thorough study of linear equations is carried out in Volume.线性微分方程的最简单
17、类型及其应用也会在介绍性的线性微分方程的最简单类型及其应用也会在介绍性的本章中讨论。线性方程的深入研究将在第二卷中进行。本章中讨论。线性方程的深入研究将在第二卷中进行。17Mathematical Englishf(x),f(x),9-B18Mathematical EnglishBy the order of an equation is meant the order of the highest derivative which appears.方程的阶指的是出现在其中的最高阶导数的阶数。方程的阶指的是出现在其中的最高阶导数的阶数。By A is meant B.A的含义为的含义为B;A
18、即即B;A指的是指的是B。19Mathematical EnglishIn this chapter we shall begin our study with first-order equations which can be solved for and written as follows.本章将开始研究通过本章将开始研究通过 可以求解可以求解的并可以写成如下形式的一阶的并可以写成如下形式的一阶方程。方程。20Mathematical Englishwhere the expression f(x,y)on the right has various special forms.A d
19、ifferentiable function y=Y(x)will be called a solution of(9.2)on an interval I if the function Y and its derivative Y satisfy the relation Y(x)=f x,Yx for every x in I.右边的函数右边的函数 f(x,y)具有不同的表达方式。称可微函数具有不同的表达方式。称可微函数y=Y(x)为一阶微分方程(为一阶微分方程(9.2)在区间)在区间I上的一个解,如果对于区间上的一个解,如果对于区间I中中的每一个的每一个x,函数及其导数都满足关系函数及
20、其导数都满足关系Y(x)=f x,Yx。The simplest case occurs when f(x,y)is independent of y.In this case,(9.2)becomes(9.3)y=Q(x).最简单的情况是最简单的情况是f(x,y)与与y无关,在这种情况下,(无关,在这种情况下,(9.2)成)成为为(9.3)y=Q(x)。21Mathematical Englishsay,where Q is assumed to be a given function defined on some interval I.To solve the differential
21、equation(9.3)means to find a primitive of Q.The Second fundamental theorem of calculus tells us how to do it when Q is continuous on an open interval I.其中其中Q为定义在某区间为定义在某区间I上的给定函数。解微分方程上的给定函数。解微分方程(9.3)即寻找即寻找Q的一个原函数。当的一个原函数。当Q在开区间在开区间I上连续时,上连续时,由微积分第二基本定理可知如何求解此方程。由微积分第二基本定理可知如何求解此方程。We simply integr
22、ate Q and add any constant.我们直接对我们直接对Q积分并加上任意常数。积分并加上任意常数。Integrate Q with respect to x.对对Q关于变量关于变量x进行积分。进行积分。22Mathematical Englishwhere C is any constant(usually called an arbitrary constant of integration).The differential equation(9.3)has infinitely many solutions,one for each value of C.其中其中C是任
23、意常数(通常被称为任意积分常数)。微分方程是任意常数(通常被称为任意积分常数)。微分方程(9.3)有无穷多解,每一个解对应一个常数)有无穷多解,每一个解对应一个常数C。()yQ x dxC Thus,every solution of(9.3)is included in the formula(9.4)因此因此,(9.3)的每一个解都包含在公式的每一个解都包含在公式(9.4)中。中。()yQ x dxC 23Mathematical EnglishIf it is not possible to evaluate the integral in(9.4)in terms of familia
24、r functions,such as polynomials,rational functions,trigonometric and inverse trigonometric functions,logarithms,and exponentials,still we consider the differential equation as having been solved if the solution can be expressed in terms of integrals of known functions.In actual practice,there are va
25、rious methods for obtaining approximate evaluations of integrals which lead to useful information about the solution.多项式,有理函数,三角函数和反三角函数,对数和指数函数多项式,有理函数,三角函数和反三角函数,对数和指数函数Automatic high-speed computing machines are often designed with this kind of problem in mind.自动高速计算机在设计时就考虑到这类问题的处理。自动高速计算机在设计时就考
26、虑到这类问题的处理。24Mathematical EnglishEXAMPLE.Linear motion determined from the velocity.Suppose a particle moves along a straight line in such a way that its velocity at time t is 2sint.Determine its position at time t.例例 直线运动的速度的确定。设质点沿直线运动的速度函数为直线运动的速度的确定。设质点沿直线运动的速度函数为2sint。试确定其在时刻。试确定其在时刻t的位置。的位置。Sol
27、ution.If Y(t)denotes the position at time t measured from some starting point,then the derivative Y(t)represents the velocity at time t.We are given that Y(t)=2sint.Integrating,we find that Y(t)=-2cos t+C.解解 令令Y(t)表示从某起点开始至时刻表示从某起点开始至时刻t的位置,则导数的位置,则导数Y(t)表示表示在时刻在时刻t的速度。由已知,的速度。由已知,Y(t)=2sint.积分即可得积分
28、即可得 Y(t)=-2cos t+C.25Mathematical EnglishThis is all we can deduce about Y(t)from a knowledge of the velocity alone;这是我们仅由速度可以推断出的所有这是我们仅由速度可以推断出的所有Y(t);some other piece of information is needed to fix the position function.We can determine C if we know the value of Y at some particular instant.For
29、example,if Y(0)=0;then C=2,and the position function is Y(t)=2-2cost.But if Y(0)=2,then C=4 and the position function is Y(t)=4-2cost.26Mathematical EnglishIn some respects the example just solved is typical of what happens in general.Some-where in the process of solving a first-order differential e
30、quation,an integration is required to remove the derivative y and in this step an arbitrary constant C appears.在解一阶微分方程的过程中,为了消去导数在解一阶微分方程的过程中,为了消去导数y,需要,需要在某一步进行积分,这时候就出现了一个任意常数。在某一步进行积分,这时候就出现了一个任意常数。27Mathematical EnglishThe way in which the arbitrary constant C enters into the solution will depe
31、nd on the nature of the given differential equation.It may appear as an additive constant,as in Equation(9.4),but it is more likely to appear in some other way.For example,when we solve the equation y=y in Section 9.3,we shall find that every solution has the form y=Cex.28Mathematical EnglishIn many
32、 problems it is necessary to select from the collection of all solutions one having a prescribed value at some point.The prescribed value is called an initial condition,and the problem of determining such a solution is called an initial-value problem.This terminology originated in mechanics where,as
33、 in the above example,the prescribed value represents the displacement at some initial time.29Mathematical English本小节重点掌握本小节重点掌握如果一个微分方程的未知函数是多元函数,则称为偏微如果一个微分方程的未知函数是多元函数,则称为偏微分方程。分方程。A differential equation is called partial differential equation if the unknown of it is a function of two or more va
34、riables.30Mathematical EnglishNew Words&Expressionscomponent 分量,成分分量,成分 matrix 矩阵矩阵dependent 相关的相关的 independent 无关的无关的 finite dimensional 有限维的有限维的 scalar 标量、数量标量、数量infinite dimensional 无限维的无限维的 n-tuple n元组元组hold trivially 显然成立显然成立 span 张成、支撑张成、支撑The Pythagorean identity 勾股定理勾股定理/毕达哥拉斯等式毕达哥拉斯等式2.10 线
35、性空间中的相关与无关集线性空间中的相关与无关集Dependent and Independent Sets in a Linear Space31Mathematical EnglishNew Words&Expressions(P90 生词与词组二生词与词组二):consistent 相容的相容的 matrix 矩阵矩阵column 列列 row 行行determinate 行列式行列式 reducible 可简化的可简化的 inverse 逆逆 matrix of coefficients 系数矩阵系数矩阵entry of matrix 矩阵的元矩阵的元 square matrix 方阵方
36、阵polynomial in x x的多项式的多项式 untenable 不可到达的不可到达的simultaneous linear equations 联立方程联立方程32Mathematical EnglishIn recent years the applications of matrices in mathematics and in many diverse fields have increased with remarkable speed.Matrix theory plays a central role in modern physics in the study of
37、quantum mechanics.近年来,在数学和许多各种不同的领域中,矩阵的应近年来,在数学和许多各种不同的领域中,矩阵的应用一直以惊人的速度不断增加。在研究量子力学时,用一直以惊人的速度不断增加。在研究量子力学时,矩阵理论在现代物理学上起着矩阵理论在现代物理学上起着主要的作用主要的作用。10-C Applications of matrices33Mathematical EnglishMatrix methods are used to solve problems in applied differential equations,specifically,in the area o
38、f aerodynamics,stress and structure analysis.One of the most powerful mathematical methods for psychological studies is factor analysis,a subject that makes wide use of matrix methods.解决应用微分方程,特别是在空气动力学,应力和结解决应用微分方程,特别是在空气动力学,应力和结构分析中的问题,要用矩阵方法。心理学研究上一种构分析中的问题,要用矩阵方法。心理学研究上一种最强有力的数学方法是因子分析,这也广泛的使用矩最
39、强有力的数学方法是因子分析,这也广泛的使用矩阵阵(方方)法法.34Mathematical EnglishRecent developments in mathematical economics and in problems of business administration have led to extensive use of matrix methods.The biological sciences,and in particular genetics,use matrix techniques to good advantage.近年来,在近年来,在数量经济学数量经济学和和企
40、业管理企业管理问题方面的发展问题方面的发展已经导致广泛的使用矩阵法。生物科学,特别在遗已经导致广泛的使用矩阵法。生物科学,特别在遗传学方面,用矩阵的技术很有成效。传学方面,用矩阵的技术很有成效。35Mathematical EnglishNo matter what the students field of major interest is,knowledge of the rudiments of matrices is likely to broaden the range of literature that he can read with understanding.不管学生主要
41、兴趣是什么,矩阵不管学生主要兴趣是什么,矩阵基本原理基本原理的知识都的知识都可能扩大他能读懂的可能扩大他能读懂的文献文献的范围。的范围。The solution of n simultaneous linear equations in n unknowns is one of the important problems of applied mathematics.解一有解一有n个未知数的个未知数的n个联立(线性)方程组是应用个联立(线性)方程组是应用数学的一个重要问题。数学的一个重要问题。36Mathematical EnglishDescartes,the inventor of an
42、alytic geometry and one of the founders of modern algebraic notation,believed that all problems could ultimately be reduced to the solution of a set of simultaneous linear equations.解析几何的发明者和现代代数计数法的创始人之一解析几何的发明者和现代代数计数法的创始人之一笛卡儿相信,所有的问题最后都能简化为解一组联笛卡儿相信,所有的问题最后都能简化为解一组联立方程。立方程。37Mathematical English
43、Although this belief is now known to be untenable,we know that a large group of significant applied problems from many different disciplines are reducible to such equations.虽然这种信念现在认为是站不住脚的,但是,我们虽然这种信念现在认为是站不住脚的,但是,我们知道,从许多不同的学科里的一大群重要的应用问知道,从许多不同的学科里的一大群重要的应用问题都可以约简为这类的方程。题都可以约简为这类的方程。38Mathematica
44、l EnglishMany of the applications,require the solution of a large number of simultaneous linear equations,sometimes in the hundreds.The advent of computers has made the matrix methods effective in the solution of these formidable problems.许多应用要求解大量的,往往数以百计的联立方程,许多应用要求解大量的,往往数以百计的联立方程,计算机的发明已经使得矩阵方法在
45、解这些计算机的发明已经使得矩阵方法在解这些难以解决难以解决的的问题方面非常活跃。问题方面非常活跃。39Mathematical EnglishFrom the above discussion,we see that the problem of solving n simultaneous linear equation in n unknowns is reduced to the problem of finding the inverse of the matrix of coefficients.(P89 下数第下数第9行行)从上面的讨论,我们看到解有从上面的讨论,我们看到解有n个未
46、知数的个未知数的n个联立个联立方程问题简化成求系数矩阵的逆矩阵的问题。方程问题简化成求系数矩阵的逆矩阵的问题。40Mathematical EnglishIt is therefore not surprising that in books on the theory of matrices the techniques of finding inverse matrices occupy considerable space.因此,在矩阵论的书中,用大量的篇幅来讲求逆矩因此,在矩阵论的书中,用大量的篇幅来讲求逆矩阵的技巧就不奇怪了。阵的技巧就不奇怪了。Of course,we will n
47、ot in our limited treatment discuss such techniques.当然,我们在这有限的叙述中不会讨论这类的技巧。当然,我们在这有限的叙述中不会讨论这类的技巧。41Mathematical EnglishNot only are matrix methods useful in solving simultaneous equations,but they are also useful in discovering whether or not the set of equations are consistent,in the sense that th
48、ey lead to solutions,and in discovering whether or not the set of equations are determinate,in the sense that they lead to unique solution.矩阵方法不仅在解联立方程中有用,而且在发现方矩阵方法不仅在解联立方程中有用,而且在发现方程组是否程组是否相容相容,即方程组是否有解的问题,以及方,即方程组是否有解的问题,以及方程组是否是程组是否是确定确定的,即是否有惟一解等方面,都是的,即是否有惟一解等方面,都是有用的。有用的。42Mathematical Englis
49、h作业:作业:P78:1,2(1)P87:1,2(1)43Mathematical EnglishNew Words&Expressions:assert 断言,主张断言,主张 predicate 谓词谓词 conjunction 合取合取 quantifier 量词量词connective 连词连词 quantification 量词化量词化 disjunction 析取析取 statement 语句语句2.11 数理逻辑入门数理逻辑入门Elementary Mathematical Logic44Mathematical EnglishKey points:45Mathematical E
50、nglish 1.了解谓词和量词的基本表示方法。了解谓词和量词的基本表示方法。2.掌握概率论基本的表示方法。掌握概率论基本的表示方法。46Mathematical EnglishStatements involving variables,such as“x3”,“x+y=3”,“x+y=z”are often found in mathematical assertions and in computer programs.11-A Predicates包含变量的语句,比如包含变量的语句,比如“x3”,“x+y=3”,“x+y=z”常出现在数常出现在数学论断和计算机程序中。学论断和计算机程序