Re: [理工] [工數]-二階變係數ode
※ 引述《mdpming (★ｐｉｇｍｉｎｇ★)》之銘言：
: 但是我寫寫真密集題目
: 遇到一題 xy'' +2(x-1)y' + (x-2)y = 0
: mx
: 是 try y = e
: n
: 用 y = cx
: 做不出來也
...
原文吃光光
看不懂您想問啥 ...
不過解2階非線性的 O.D.E. (並非指此題是 non-linear)
可以提供你一個方法：
define D = d/dx
then xy'' +2(x-1)y' + (x-2)y = 0
→ [xD^2 + 2(x-1)D + (x-2)]y = 0
→ (xD + x-2)(D + 1)y = 0
令 u(x) = (D + 1)y = y' + y
則原題目可變成解以下聯立方成組：
xu' + (x-2)u = 0 _____(1)
u = y' + y _____(2)
from (1) : 1/u du = (2/x - 1) dx
→ ln|u| = 2ln|x| - x + ln(C1)
or u = C1*x^2*e^(-x)
from (2) and by (1) :
C1*x^2*e^(-x) = y' + y
→ y*e^x = ∫C1*x^2 dx
= C1*x^3/3 + C2
or y = C3*x^3*e^(-x) + C2*e^(-x) where C3 = C1/3
ps: 若 xy'' +2(x-1)y' + (x-2)y = f(x) 也能這樣解
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◆ From: 140.113.141.151
※ 編輯: doom8199 來自: 140.113.141.151 (08/28 17:29)
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※ 編輯: doom8199 來自: 140.113.141.151 (08/29 00:16)
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