Re: 高階非線性ODE

看板Math作者ntust661 (TOEFL_5!)時間5年前 (2018/08/16 23:28)推噓2(2推 0噓 3→)

留言5則, 2人參與討論串2/2 (看更多)

※ 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) === -- ※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 123.193.227.179 ※ 文章網址: https://www.ptt.cc/bbs/Math/M.1534433290.A.624.html

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nini851216

08/17 11:08, 5年前 , 1^F

08/17 11:08, 1^F