数学专业英语－First Order Differential Equations

yunjiang · 发表于 2004-5-6 09:30:47

<

>A differential equation is an equation between specified derivatives of a function, its
<

>valves,and known quantities.Many laws of physics are most simply and naturally formu-
<

>lated as differential equations (or DE’s, as we shall write for short).For this reason,DE’s 
have been studies by the greatest mathematicians and mathematical physicists since the 
time of Newton..
Ordinary differential equations are DE’s whose unknowns are functions of a single va-
riable;they arise most commonly in the study of dynamic systems and electric networks.
They are much easier to treat than partial differential equations,whose unknown functions
depend on two or more independent variables.
Ordinary DE’s are classified according to their order. The order of a DE is defined as 
the largest positive integer, n, for which an n-th derivative occurs in the equation. This
chapter will be restricted to real first order DE’s of the form
 Φ(x, y, y′)=0 (1)
Given the function Φof three real variables, the problem is to determine all real functions y=f(x) which satisfy the DE, that is ,all solutions of(1)in the following sense.
DEFINITION A solution of (1)is a differentiable function f(x) such that 
Φ(x. f(x),f′(x))=0 for all x in the interval where f(x) is defined.
EXAMPLE 1. In the first-other DE
 x+yy′=0 (2)
the function Φ is a polynomial function Φ(x, y, z)=x+ yz of three variables in-
volved. The solutions of (2) can be found by considering the identity
d(x²+y²)/d x=2(x+yyˊ).From this identity,one sees that x²+y² is a con-
stant if y=f(x) is any solution of (2).
The equation x²+y²=c defines y implicitly as a two-valued function of x,
for any positive constant c.Solving for y,we get two solutions,the(single-valued)
functions y=±(c-x²)0.5 ,for each positive constant c.The graphs of these so-
lutions,the so-called solution curves,form two families of scmicircles,which fill the upper half-plane y>0 and the lower half-plane y>0,respectively.
On the x-axis,where y=0,the DE(2) implies that x=0.Hence the DE has no solutions
which cross the x-axis,except possibly at the origin.This fact is easily overlooked,
because the solution curves appear to cross the x-axis;hence yˊdoes not exist,and the DE (2) is not satisfied there.
The preceding difficulty also arises if one tries to solve the DE(2)for yˊ. Dividing through by y,one gets yˊ=-x/y,an equation which cannot be satisfied if y=0.The preceding difficulty is thus avoided if one restricts attention to regions where the DE(1) is normal,in the following sense.
 DEFINITION. A normal first-order DE is one of the form
 yˊ=F(x,y) (3)
In the normal form yˊ=-x/y of the DE (2),the function F(x,y) is continuous in the upper half-plane y>0 and in the lower half-plane where y<0;it is undefined on the x-axis.
 
Fundamental Theorem of the Calculus.
 
The most familiar class of differential equations consists of the first-order DE’s of the form 
 yˊ=g(x) (4)
Such DE’s are normal and their solutions are descried by the fundamental thorem of the calculus,which reads as follows.
FUNDAMENTAL THEOREM OF THE CALCULUS. Let the function g(x)in DE(4) be continuous in the interval a<x<b.Given a number c,there is one and only one solution f(x) of the DE(4) in the interval such that f(a)=c. This solution is given by the definite integral
f(x)=c+∫axg(t)dt , c=f(a) (5)
This basic result serves as a model of rigorous formulation in several respects. First,it specifies the region under consideration,as a vertical strip a<x
We recall that the definite integral 
∫axg(t)dt=lim(maxΔtk->0)Σg(tk)Δtk , Δtk=tk-tk-1 (5ˊ)
is defined for each fixed x as a limit of Ricmann sums; it is not necessary to find a formal expression for the indefinite integral ∫ g(x) dx to give meaning to the definite integral ∫axg(t)dt,provided only that g(t) is continuous.Such functions as the error function crf x =(2/(π)0.5)∫0xe-t² dt and the sine integral function SI(x)=∫x∞[(sin t )/t]dt are indeed commonly defined as definite integrals.
 
Solutions and Integrals
 
According to the definition given above a solution of a DE is always a function. For example, the solutions of the DE x+yyˊ=0 in Example I are the functions y=± (c-x²)0.5,whose graphs are semicircles of arbitrary diameter,centered at the origin.The graph of the solution curves are ,however,more easily described by the equation x²+y²=c,describing a family of circles centered at the origin.In what sense can such a family of curves be considered as a solution of the DE ?To answer this question,we require a new notion.
DEFINITION. An integral of DE(1)is a function of two variables,u(x,y),which assumes a constant value whenever the variable y is replaced by a solution y=f(x) of the DE.
In the above example, the function u(x,y)=x²+y² is an integral of the DE x+yyˊ =0,because,upon replacing the variable y by any function ±( c-x²)0.5,we obtain u(x,y)=c.
The second-order DE
 d²x/dt²=-x (2ˊ)
becomes a first-order DE equivalent to (2) after setting dx/dx=y:
y ( dy/dx )=-x (2)
As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE.When the DE (2ˊ)
is interpreted as equation of motion under Newton’s second law,the integrals 
c=x²+y² represent curves of constant energy c.This illustrates an important principle:an integral of a DE representing some kind of motion is a quantity that remains unchanged through the motion.

yunjiang · 发表于 2004-5-6 09:31:03

<

0cm 0cm 0pt; TEXT-ALIGN: center" align=center>Vocabulary<

0cm 0cm 0pt"> <

0cm 0cm 0pt">differential equation 微分方程 error function 误差函数 ordinary differential equation 常微分方程 sine integral function 正弦积分函数order 阶,序 diameter 直径derivative 导数 curve 曲线known quantities 已知量 replace 替代unknown 未知量 substitute 代入single variable 单变量 strip 带形dynamic system 动力系统 exact differential 恰当微分electric network 电子网络 line integral 线积分partial differential equation 偏微分方程 path of integral 积分路径classify 分类 endpoints 端点polynomial 多项式 general solution 通解several variables 多变量 parameter 参数family 族 rigorous 严格的semicircle 半圆 existence 存在性half-plane 半平面 initial condition 初始条件region 区域 uniqueness 唯一性normal 正规,正常 Riemann sum 犁曼加identity 恒等(式)

yunjiang · 发表于 2004-5-6 09:31:20

<

0cm 0cm 0pt; TEXT-ALIGN: center" align=center>Notes<

0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; tab-stops: list 18.0pt; mso-list: l29 level1 lfo19">1. The order of a DE is defined as the largest positive integral n,for which an nth derivative occurs in the question.<

0cm 0cm 0pt"> 这是另一种定义句型,请参看附录IV.此外要注意nth derivative 之前用an 不用a .2. This chapter will be restricted to real first order differential equations of the form Φ(x,y,yˊ)=0意思是;文章限于讨论形如 Φ(x,y,yˊ)=0的实一阶微分方程.有时可以用of the type代替 of the form 的用法. The equation can be rewritten in the form yˊ=F(x,y).3. Dividing through by y,one gets yˊ=-x/y,… 划线短语意思是:全式除以y4. As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE 这里x²+y²=c 因c是参数,故此方程代表一族曲线,由此”曲线”这一词要用复数curves.5. Their solutions are described by the fundamental theorem of the calculus,which reads as follows. 意思是:它们的解由微积分基本定理所描述,(基本定理)可写出如下. 句中reads as follows 就是”写成(读成)下面的样子”的意思.注意follows一词中的”s”不能省略.

yunjiang · 发表于 2004-5-6 09:31:32

<

0cm 0cm 0pt; TEXT-ALIGN: center" align=center>Exercise<

0cm 0cm 0pt">Ⅰ.Translate the following passages into Chinese:<

0cm 0cm 0pt"> 1.A differential M(x,y) dx +N(x,y) dy ,where M, N are real functions of two variables x and y, is called exact in a domain D when the line integral ∫c M(x,y) dx +N(x,y) dy is the same for all paths of integration c in D, which have the same endpoints.Mdx+Ndy is exact if and only if there exists a continuously differentiable function u(x,y) such that M= u/ x, N=u/ y.2. For any normal first order DE yˊ=F(x,y) and any initial x0 , the initial valve problem consists of finding the solution or solutions of the DE ,for x>x0 which assumes a given initial valve f(x0)=c.3. To show that the initial valve problem is well-set requires proving theorems of existence (there is a solution), uniqueness (there is only one solution) and continuity (the solution depends continuously on the initial value). Ⅱ. Translate the following sentences into English:1) 因为y=ч(x) 是微分方程dy/ dx=f(x,y)的解,故有dч(x)/dx=f (x,ч(x))2) 两边从x0到x取定积分得ч(x)-ч(x0)=∫x0x f(x,ч(x)) dx x0<x<x0+h3) 把y0=ч(x0)代入上式, 即有ч(x)=y0+∫x0x f(x,ч(x)) dx x0<x<x0+h4) 因此 y=ч(x) 是积分方程 y=y0+∫x0x f (x,y) dx 定义于x0<x<x0+h 的连续解. Ⅲ. Translate the following sentences into English:1) 现在讨论型如 y=f (x,yˊ) 的微分方程的解,这里假设函数 f (x, dy/dx) 有连续的偏导数.2) 引入参数dy/dx=p, 则已给方程变为 y=f (x,p).3) 在 y=f (x,p) x p=dy/dx p= f/ x+f/ p dp/dx4) 这是一个关于x和p的一阶微分方程,它的解法我们已经知道.5) 若(A)的通解的形式为p=ч(x,c) ,则原方程的通解为y=f (x,ч(x,c)).6) 若(A) 有型如x=ψ(x,c)的通解,则原方程有参数形式的通解 x=ψ(p,c) y=f(ψ(p,c)p) 其中p是参数,c是任意常数.

		自动登录	找回密码
密码			注-册-帐-号