Re: [其他] 一階ODE

[Honor1984 (奈何上天造化弄人？)]

※ 引述《S1010630 (ggod)》之銘言：
: https://i.imgur.com/8K0px28.jpg
: 請問第一小題該如何解
: 解答贈微薄p幣 感謝...

(2x + y)y' = (4/y)x^2 + y + 4x

Set u = 2x + y

y' = u' - 2

u[u' - 2] = 4x^2 /[u - 2x] + u + 2x   or u = 0  即y = -2x

u' - 2 = 4x^2 /[u(u - 2x)] + 1 + 2x/u

       = 2x[1/(u - 2x) - 1/u] + 1 + 2x/u

       = 2x/(u - 2x) + 1

       = u/(u - 2x)

u' = (3u - 4x)/(u - 2x)

齊次ODE

v = u/x代入解

xv' + v = (3v - 4)/(v - 2)

=> xv' = (-v^2 + 5v - 4)/(v - 2) = -(v - 1)(v - 4)/(v - 2)

=> (1/x)dx = [-(1/3)/(v - 1) - (2/3)/(v - 4)]dv

=> c + lnx = (-1/3)ln|v - 1| - (2/3)ln|v - 4|

=> c + lnx = -(1/3)ln|(2x + y)/x - 1| - (2/3)ln|(2x + y)/x - 4|

=> Ax = x(x + y)^(1/3) (y - 2x)^(2/3)

=> C = (x + y) (y - 2x)^2

加上最前面or y + 2x = 0

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→ Vulpix      : +1不見了。
→ Vulpix      : 這題答案應該是y=-2x或(y+x)(y-2x)^2=C
謝謝提醒，我漏掉了1
已修正
※ 編輯: Honor1984 (111.243.61.160), 10/24/2018 23:14:27