differential equation

2009-01-26 7:42 pm
根據 → http://hk.knowledge.yahoo.com/question/question?qid=7009012600122

(1): A'(x) cos kx + B'(x) sin kx = 0
(2): -k A'(x) sin kx + k B'(x) cos kx = f(x)

以上既set up點黎 , 請解釋.

回答 (1)

2009-01-27 1:57 am
✔ 最佳答案
It's just a method
In order to use the method of variation of parameters we need to know that
圖片參考:http://www.sosmath.com/diffeq/second/variation/img3.gif
. The idea behind the method of variation of parameters is to look for a particular solution such as

圖片參考:http://www.sosmath.com/diffeq/second/variation/img4.gif

where
圖片參考:http://www.sosmath.com/diffeq/second/variation/img6.gif
are functions. From this, the method got its name.
The functions
圖片參考:http://www.sosmath.com/diffeq/second/variation/img6.gif
are solutions to the system

圖片參考:http://www.sosmath.com/diffeq/second/variation/img7.gif
,
There are some tricks behind it and you can find the reasons in most of diferential equation books
which implies

圖片參考:http://www.sosmath.com/diffeq/second/variation/img8.gif
,
where
圖片參考:http://www.sosmath.com/diffeq/second/variation/fonction2.gif
. Therefore, we have

圖片參考:http://www.sosmath.com/diffeq/second/variation/img9.gif


Summary:Let us summarize the steps to follow in applying this method:


(1)
Find
圖片參考:http://www.sosmath.com/diffeq/second/variation/img2.gif
a set of fundamental solutions of the associated homogeneous equation

y'' + p(x)y' + q(x)y = 0.;

(2)
Write down the form of the particular solution

圖片參考:http://www.sosmath.com/diffeq/second/variation/img10.gif
;

(3)
Write down the system

圖片參考:http://www.sosmath.com/diffeq/second/variation/img7.gif
;

(4)
Solve it. That is, find
圖片參考:http://www.sosmath.com/diffeq/second/variation/img6.gif
;
(5)
Plug
圖片參考:http://www.sosmath.com/diffeq/second/variation/img6.gif
into the equation giving the particular solution.
Now for the question y'' + k^2 y = f(x), the set of fundamental solutions of the equation y'' + ky = 0 is coskx,sinkx ;
(2)
The particular solution is given as
yp= A(x) cos kx + B(x) sin kx, where A, B are functions.





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