Re: [理工] [工數] ODE
※ 引述《kwenchou (活老百姓)》之銘言：
: 這裡有幾題ODE請各位板友幫個忙了...
: ╴╴ -1
: (1) (x+3y)dx-(x-y)dy = 0 with y(0)=1 , ㏑(√x+y ) = ?
: ╴╴ -1 1 x
: 解答: ㏑(√x+y ) = -(——)㏑(x+y) = ——
: 2 x+y
: 這一題我首先用齊次下去算 可是怎麼都湊不出他要的東西...
我只會解ODE...
令 u = x + y , du = dx + dy 帶回
(x + 3u - 3x)dx - (x - u + x)(du - dx) = 0
2udx - 2xdu + udu = 0
2 x
2 u d(---) + udu = 0
u
x 1
2 d(---) + ---du = 0
u u
x
2 --- + ln|u| = c
u
x
2 ------- + ln|x + y| = c ,帶入y(0) = 1 可得c = 0
x + y
-1 x
故 ----ln|x + y| = -------
2 x + y
--
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