1、New Words & Expressions:approximate evaluation 近似估計 initial 初始的disintegrate 解體，衰變 integrate 求積分 differentiable 可微的 polynomial 多項式exponential 指數的 rational function 有理函數2.9 微分方程簡介Introduction to Differential EquationsKey points: Introduction to Differential EquationsDifficult points: Applications of m

2、atricesRequirements: 1. 理解微分方程的分類。 2. 理解矩陣學習的重要性。A large variety of scientific problems arise in which one tries to determine something from its rate of change.9-A Introduction大量的科學問題需要人們根據事物的變化率來確定該事物。For example , we could try to compute the position of a moving particle from a knowledge of its ve

3、locity or acceleration.例如，我們可以由已知速度或者加速度來計算移動質點的位置.Or a radioactive substance may be disintegrating at a known rate and we may be required to determine the amount of material present after a given time.又如，某種放射性物質可能正在以已知的速度進行衰變，需要我們確定在給定的時間后遺留物質的總量。In examples like these, we are trying to determine a

4、n unknown function from prescribed information expressed in the form of an equation involving at least one of the derivatives of the unknown function .在類似的例子中，我們力求由方程的形式表述的信息來確定未知函數，而這種方程至少包含了未知函數的一個導數。These equations are called differential equations, and their study forms one of the most challengi

5、ng branches of mathematics.這些方程稱為微分方程，對其研究形成了數學中最具有挑戰性的一個分支。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 more variables.微分方程根據未知量是單變量函數還是多變量函數分成兩個主題：常微分方程和偏微分方程。 A simple example of

6、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 .常微分方程的一個簡單例子是f(x)=f(x) ，特別地，指數函數f(x)=ex 滿足這個等式。We shall see presently that every solution of (9.1) must be of the form f(x)=Cex , where C may be any constant.我們馬上就

7、會發現(9.1)的每一個解都一定是f(x)=Cex這種形式，這里C可以是任何常數。On 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個例子。This particular one, is called Laplaces equation, appears in the theory of electricity and magnetism, fluid mechanics, and elsewhere.這個特殊的方程叫做拉普拉斯方程，出現于電磁

8、學理論、流體力學理論以及其他理論中。The study of differential equations is one part of mathematics that, perhaps more than any other, has been directly inspired by mechanics, astronomy, and mathematical physics.微分方程的研究是數學的一部分，也許比其他分支更多的直接受到力學，天文學和數學物理的推動。Its history began in the 17th century when Newton, Leibniz, and

9、 the Bernoullis solved some simple differential equations arising from problems in geometry and mechanics.微分方程起源于17世紀，當時牛頓，萊布尼茨，伯努利家族解決了一些來自幾何和力學的簡單的微分方程。These early discoveries, beginning about 1690, gradually led to the development of a lot of “special tricks” for solving certain special kinds of

10、differential equation.開始于1690年的早期發現，逐漸導致了解某些特殊類型的微分方程的大量特殊技巧的發展。 Although these special tricks are 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. 盡管這些特殊的技巧只是適用于相對較少的幾種情況，但他們能夠

11、解決許多出現于力學和幾何中的微分方程，因此，他們的研究具有重要的實際應用。Some of these special methods and some of the problems which they help us solve are discussed near the end of this chapter.這些特殊的技巧和利用這些技巧可以解決的一些問題將在本章最后討論。本小節重點掌握如果一個微分方程的未知函數是多元函數，則稱為偏微分方程。A differential equation is called partial differential equation if the un

12、known of it is a function of two or more variables.New Words & Expressions(P90 生詞與詞組二):consistent 相容的 matrix 矩陣column 列 reducible 可簡化的 determinate 行列式 row 行inverse 逆 simultaneous linear equations 聯立方程2.10 線性空間中的相關與無關集Dependent and Independent Sets in a Linear SpaceIn recent years the applications of

13、 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 quantum mechanics.近年來，在數學和許多各種不同的領域中，矩陣的應用一直以驚人的速度不斷增加。在研究量子力學時，矩陣理論在現代物理學上起著主要的作用。Matrix methods are used to solve problems in applied diffe

14、rential equations, specifically, in the area of 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. 解決應用微分方程，特別是在空氣動力學，應力和結構分析中的問題，要用矩陣方法。心理學研究上一種最強有力的數學方法是因子分析，這也廣泛的

15、使用矩陣(方)法 .Recent 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. 近年來，在數量經濟學和企業管理問題方面的發展已經導致廣泛的使用矩陣法。生物科學，特別在遺傳學方面，用矩陣的技術很有成效。No

16、 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 .不管學生主要興趣是什么，矩陣基本原理的知識都可能擴大他能讀懂的文獻的范圍。The solution of n simultaneous linear equations in n unknowns is one of the important p

17、roblems of applied mathematics. 解一有n個未知數的n個聯立（線性）方程組是應用數學的一個重要問題。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. 解析幾何的發明者和現代代數計數法的創始人之一

18、笛卡兒相信，所有的問題最后都能簡化為解一組聯立方程。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. 雖然這種信念現在認為是站不住腳的，但是，我們知道，從許多不同的學科里的一大群重要的應用問題都可以約簡為這類的方程。Many of the applications, require the sol

19、ution 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. 許多應用要求解大量的，往往數以百計的聯立方程，計算機的發明已經使得矩陣方法在解這些難以解決的問題方面非常活躍。From 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 下數第9行)從上面的討論，我們看到解有n個未知數的n個聯立方程問題簡化成求系數矩陣的逆矩陣的問題。It is therefore not surprising that