数学专业英语论文

课文9-B Terminology and notation

when we work with a differential equation such as(9.1),it is customary to write y in place of f（x) and y' in place of f'(x)，the higher derivatives being denoted by y"，y'''，etc.Of course ，other letters such as u，v，z，etc.are also used instead of y. By the order of an equation is meant the order of the highest derivatives which appears.For example ,(9.1)is first-order equation which may be written as y'=y.The differential equation

)

sin(xy"

y

x

y'3+

=is one of second order.

In this chapter we shall begin our study with firs-order equations which can be solved for y' and written as follows:

(9.2) y'=f(x,y),

Where the expression f(x,y) on the right has various special forms. A defferentiable function y=Y(x) will be called a solution of (9.

2) on an interval I if the function Y and and its derivative Y' satisfy the relation

Y'=f[x,Y(x)]

For every x in I. The simplest case occurs when f(x,y)is independent of y.In this case , (9.2) becomes

(9.3) y'=Q(x),

Say, where Q is assumed to be a liven function defined on some interval I. To solve the differential 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. We simply integrate Q and add any constant.Thus，every solution of (9.3) is included in the formula

（9.4）y=∫Q(x)dx + C,

where C is any constant ( usually called an arbitrary constant of integration). The differential equation(9.3) has infinitely many 课文9—B 术语和符号

当我们在求解像(9.1)式的微分方程时，习惯用y代替f(x)，用y’代替f'(x)，用高阶导数y''和y'''等表示。当然，其他的字母如u,v,z等等，同样可以用来代替y。微分方程和阶数指的是现在其中的高阶导数的阶。例如，（9.1）式是一个一次方程可以写成y'=y。

微分方程

)

s i n(x y"

y

x

y'3+

=是一个二阶的。

在这章我们将会学习到可以求解y'的一阶微分方程。一阶方程可以被写成这样：（9.2）y'=f(x,y),

其中，右边有各个特殊形式表示。如果对于区间I中的每一个x函数y和他的倒数满足

Y'=f[x,Y(x)]

那么可微函数就为（9.2）在区间I中的一个解，最简单的形式是f（x，y）与y无关。在这种情况下，（9.2）式变成了

（9.3）y'=Q(x),

表明，其中Q是假定在区间中的一个给定函数，对于一个给定的函数定义在各个区间I.求解微分方程（9.3）就意味着找到原始的区间Q。第二基本积分定理告诉我们，当Q位于一个连续的开放的区间I 时该怎么做。我们直接对Q积分并加上任意常数。因此，y=∫Q(x)dx + C包含了（9.3）式的所有解

（9.4）y=∫Q(x)dx + C,

其中C为任意常数（通常被称为积分下限的任意常数），微分方程（9.3）有无穷多个解，每个解对应一个C。

solutions, one for each value of C.

If it is not possible to evaluate the integral in(9. 4)in terms of familiar function,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 various methods for obtaining approximate evaluations of integrals which lead to useful information about the solution. Automatic high-speed computing machines are often designed with this kind of problem in mind .

Example. 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 2sin t.Determine its position at time t..

Solution. if Y(t) denotes the position at time t measured from some staring point,then the derivative Y'(t) represents the velocity at time t. We are given that

Y'(t)=2sin t. Integrating，we find that

Y(t)=2∫sin t dt+C=2cos t +C.

This is all we can deduce about Y(t) from a knowledge of the velocity alone;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 example，if Y( 0)=0，then C=2 and the position function is Y( t )=2-2cos t. But if Y( 0)=2，then C=4 and the position function is Y( t)=4-2cos t.

In some respects the example just solved is typical of what happens in general.Some-where in the process of solving a first-order differential equation，an integration is required to remove the derivative y' and in this step an arbitrary

如果这不是在一些常见条件下的函数对（9.4）求积分，如多项式，有理方程，三角函数和反三角函数，对数和指数函数，我们仍然要把微分方程看作是一个被解答的函数，如果这个函数可以表示成已知的函数的话。在实际的练习中，有很多方法可以获得

近似值，这些值可以给正确答案提供很多有用的信息。自动高速运行的计算机在设计时就常常考虑到对这类问题的处理。

例如：直线运动取决于速度。假设一个质点按照这种方法沿着一条直线运动,它的速度在t时刻是 2sin t。在t时刻确定该质点位置。

结论：如果Y(t)表示被测点在t时刻的位置，那么，Y'(t)表示在t时刻的速率。我们便可得出以下结论：

Y'(t)=2sin t.

综上所述：

Y(t)=2∫sin t dt+C=2cos t +C.

这就是所有我们能从速度推断出的关于Y(t)的知识；其他的一些信息需要从位置函数中整理得到。在一些特殊的时刻，如果我们知道Y值，便可推断出C。例如，如果Y(0)=2，那么C=4，并且位置函数是：

Y( t)=4-2cos t.

在某些方面，刚才例题所解决的问题是一般情况下都可行。在求解一阶微分方程的过程中，为了消除导数y'，需要进一步进行积分，这时候就出现了一个任意常数C。将常数C

constant C appears. The way in which the arbitrary constant C enters into the solution will depend 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=Ce^x.

In many 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 in the above example,the prescribed value represents the displacement at some initial time.

熊进学号：20083729 代入方程的解决方法依赖于所给常微分方程。

我们就会发现，所有的解都有这样的形式

y=Ce^x.

在很多问题中，必须从得出的所有解中选择一个，在这个解的某种意义上满足所给定的值。这个给定的值叫做初始条件，这种给定了初始条件的问题叫做初始问题。这个术语来源于力学，在上面的列子中，规定的值代表最初的位移量。