Re: [理工] [工數]-一階ODE
※ 引述《doom8199 (～口卡口卡 修～)》之銘言：
: 3 2 2
: (x y - 1)dx + x dy = 0
: 題目看起來短短的
: 可是我試過很多方法都解不太出來 QQ
: 這個 ODE 是我在算一個東西的時候化簡得到的
: 所以手邊沒答案
: 麻煩大家了 > <
3 2 2
(x y - 1)dx + x dy = 0
y'=-xy^2+x^-2
let y=p'/(px)
y'=p''(xp)^-1 - p'^2p^-2x^-1 - p'p^-1x^-2
=-p'^2p^-2x^-1+x^-2
=>p''-p'x^-1=x^-1p
=>x^2p''-xp'-xp=0
let t=x^1/2
dp dp dt 1 dp
--= --- ---=---x^-(1/2)----
dx dt dx 2 dt
d^2p -x^-(3/2) dp x^-1 d^2p
----=---------- ---- + ------ ----
dx^2 4 dt 4 dt^2
=>
-t dp t^2 d^2p t dp
---- ---- + ------ ---- - --- --- -t^2p =0
4 dt 4 dt^2 2 dt
d^2p dp
t^2---- - 3t---- -4t^2p=0
dt^2 dt
let p=vt^2
p`=v`t^2+2tv
p``=v``t^2+4tv`+2v
t^2(v``t^2+4tv`+2v)-3t(v`t^2+2tv)-4t^2(vt^2)=0
t^2v``+(4t-3t)(v`)-(6-2+4t^2)v=0
t^2v``+tv`-(4t^2+4)v=0 , modified bessel eq.
v=AI (2t)+BK (2t)
2 2
p=x(AI (2x^1/2)+BK (2x^1/2))
2 2
y=p'/(px)
我算不下去了orz........
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