Re: [理工] [工數]-ODE
※ 引述《jay0748 (山豬)》之銘言：
: dy 3y+3x^2 y^2
: — = ─────
: dx 2x^3 y-3x
: 不知如何下手
: 麻煩版友幫忙解答 謝謝
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(2x^3y - 3x)dy = (3y + 3x^2y^2) dx
→ x^3 d(y^2) - 3xdy = 3ydx + y^2 d(x^3)
→ x^6 d(y^2/x^3) = 3 d(xy) ____(1)
(假設)
→ (y^2/x^3)^m d(y^2/x^3) = 3(xy)^n d(xy) ____(2)
(y^2/x^3)^(m+1) 3(xy)^(n+1)
→ _______________ = ___________ + C
m+1 n+1
由 (1) (2) 可知:
┌ 2m - n = 0 → ┌ m = (-6/5)
└ -3m - n = 6 └ n = (-12/5)
即通解為:
(y^2/x^3)^(-1/5) 3(xy)^(-7/5)
________________ = ____________ + C
(-1/5) (-7/5)
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或是假設積分因子為 I(x,y) = (x^m)(y^n) 帶入解出 m、n 也可以
我解出來是 m = n = (-12/5)
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◆ From: 140.113.141.151
推
11/05 11:40, , 1F
11/05 11:40, 1F
x^3 d(y^2) - 3xdy = 3ydx + y^2 d(x^3)
^^^^^^ ^^^^^^^^^^^^
└─→移項 │
移項 ←──┘
→ x^3 d(y^2) - y^2 d(x^3) = 3ydx + 3xdy
^^^^^^^^^^^^^^^^^^^^^^^ vdu - udv
(合併) ___________ using d(u/v) = _________
v^2
→ x^6 d(y^2/x^3) = 3ydx + 3xdy
^^^^^^^^^^^
(合併) ________ using d(uv) = vdu + udv
→ x^6 d(y^2/x^3) = 3 d(xy)
※ 編輯: doom8199 來自: 140.113.141.151 (11/05 12:44)
→
11/05 14:00, , 2F
11/05 14:00, 2F
→
11/05 15:15, , 3F
11/05 15:15, 3F
討論串 (同標題文章)