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1、Key points:Introduction to Differential Equations Introduction to Differential EquationsDifficult points:Applications of matrices Applications of matrices第1页/共27页Requirements:Requirements:1.理解微分方程的分类。2.理解矩阵学习的重要性。第2页/共27页A large variety of scientific problems arise in which one tries to determine so
2、mething 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点的位置.第3页/共27页Or a radioactive substance may be disintegrating at a known rate and we
3、 may be required to determine 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变，需要我们确定在给定的时间后遗留物质的总量。第4页/共27页In examples like these,we are trying to determine an unknown function from prescribed information expressed in the form of an equation involving at least one of the d
4、erivatives 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。第5页/共27页These equations are called differential equations,and their study forms one of the most challenging branches of mathematics.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。Differential equations are classified under two m
5、ain headings:ordinary and partial,depending on whether the unknown is a function of just one variable or of two 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成两个主题：常微分方程和偏微分方程。第6页/共27页A simple example of an ordinary differential equation is the relation f(x)=f(x)(9.1)which is satisfied,in particular by the
6、exponential function,f(x)=ex.常微分方程的一个简单例子是f(x)=f(x)，特别地，指数函数f(x)=ex 满足这个等式。We shall see presently that every solution of(9.1)must be of the form f(x)=Cex,where C may be any constant.我们马上就会发现(9.1)的每一个解都一定是f(x)=Cex这种形式，这里C可以是任何常数。第7页/共27页On the other hand,an equation likeis an example of a partial dif
7、ferential equation.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，出现于电磁学理论、流体力学理论以及其他理论中。第8页/共27页The study of differential equations is one part of mathematics that,perhaps more tha
8、n 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 Bernoullis solved some simple differential equations arising from problems in geometry and mechanics.微分
9、方程起源于17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。第9页/共27页These early discoveries,beginning about 1690,gradually led to the development of a lot of“special tricks”for solving certain special kinds of differential equation.开始于1690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。第10页/共27页Although these special tricks a
10、re applicable in relatively few cases,they do enable us to solve many differential equations 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，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。第11页/共27页Some of these special methods and some of the problem
11、s which 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将在本章最后讨论。第12页/共27页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，则称为偏微分方程。A differential equation is called partial differential equation if the unknown of it is a function of two or more variables.第13页/共27页New Words&Expressions
12、(P90 生词与词组二):consistent 相容的 matrix 矩阵column 列 reducible 可简化的 determinate 行列式 row 行inverse 逆 simultaneous linear equations 联立方程2.10 线性空间中的相关与无关集Dependent and Independent Sets in a Linear Space第14页/共27页In recent years the applications of matrices in mathematics and in many diverse fields have increase
13、d with remarkable speed.Matrix theory plays a central role in modern physics in the study of quantum mechanics.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，矩阵理论在现代物理学上起着主要的作用。第15页/共27页Matrix methods are used to solve problems in applied differential equations,specifically,in the area of aerodynamic
14、s,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，这也广泛的使用矩阵(方)法.第16页/共27页Recent developments in mathematical economics
15、 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，在数量经济学和企业管理问题方面的发展已经导致广泛的使用矩阵法。生物科学，特别在遗传学方面，用矩阵的技术很有成效。第17页/共27页No matter what the students field of major interest
16、is,knowledge of the rudiments of matrices is likely to broaden the range of literature that he can read with understanding.不管学生主要兴趣是什么，矩阵基本原理的知识都可能扩大他能读懂的文献的范围。The solution of n simultaneous linear equations in n unknowns is one of the important problems of applied mathematics.解一有n个未知数的n个联立（线性）方程组是应
17、用数学的一个重要问题。第18页/共27页Descartes,the inventor of analytic 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笛卡儿相信，所有的问题最后都能简化为解一组联立方程。第19页/共27页Although this
18、 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。第20页/共27页Many of the applications,require the solution of a large number of simulta
19、neous 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，计算机的发明已经使得矩阵方法在解这些难以解决的问题方面非常活跃。第21页/共27页From the above discussion,we see that the problem of solving n simultaneous line
20、ar equation in n unknowns is reduced to the problem of finding the inverse of the matrix of coefficients.(P89 下数第9行)从上面的讨论，我们看到解有n个未知数的n个联立方程问题简化成求系数矩阵的逆矩阵的问题。第22页/共27页It is therefore not surprising that in books on the theory of matrices the techniques of finding inverse matrices occupy considerabl
21、e space.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。Of course,we will not 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。第23页/共27页Not only are matrix methods useful in solving simultaneous equations,but they are also useful in discovering whether or not the set of equations are consiste
22、nt,in the sense that they 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，而且在发现方程组是否相容，即方程组是否有解的问题，以及方程组是否是确定的，即是否有惟一解等方面，都是有用的。第24页/共27页作业：P78 2:(1)P87 2:(3)第25页/共27页谢 谢！第26页/共27页感谢您的观看！第27页/共27页