推 nini851216 : 我的v算出來是C2[ ln(x)+C1 ]^2 也對嗎？ 08/17 12:12
→ ntust661 : 我算錯了XD 08/17 18:02
→ ntust661 : 可以 你的答案也是對的 08/17 23:40
※ 編輯: ntust661 (123.193.227.179), 08/18/2018 03:57:52
→ nini851216 : 好！感謝大大 08/18 16:59
※ 引述《nini851216 (解壓縮)》之銘言：
: https://i.imgur.com/lZlwgNL.jpg
: 需要大大幫忙
: 兩種方法
: 帶進去之後就卡住了
y = vx,
=> dy/dx = v'x + v
=> d^2y/dx^2 = v''x+ 2v'
so 2x^2*(vx)*(v''x+2v')+(vx)^2=x^2(v'x+v)^2
x = e^t
t = lnx, dt/dx = 1/x, d^2t/dx^2 = -1/x^2 = -e^-2t
=> dx/dt = e^t
. -t .
=> dv/dx = dv/dt * dt/dx = dv/dt* e^-t = v e = v/x
=> d^2v/dx^2 = d/dt(dv/dt)* dt/dx * dt/dx + dv/dt * d^2t/dx^2
.. -2t . -2t .. .
= v e - v e = (v - v)/x^2
so...
3 .. . . 2 2 2 . 2
=> 2 x v ( x (v - v)/x^2 + 2v/x ) + v x = x ( v/x x + v )
.. . . 2 . 2
=> 2 v ( v - v + 2v ) + v = ( v + v )
.. . 2 . 2 . 2
=> 2 v v + 2 v v + v = v + 2 v v + v
.. . 2
=> 2 v v - v = 0
. ..
v = s, v = ds/dt = ds/dv*dv/dt = s*s'
2
2 v* s*s' = s
2 d s 1
─── = ───
s d v v
2 ln s = lnv + c1
2
v = (c1 t + c2)
2
y = x (c1 lnx + c2)
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