2題微分方程~~~高手入內

2009-12-26 8:12 am
1. y'=(y-4x+8xy^3)/(x+12x^2y^2)
2. 2x^2yy''+y^2=x^2(y')^2

回答 (3)

2010-01-02 4:35 pm
✔ 最佳答案
2.

2x²yy'' + y² = x²(y')²
yy'' + y²/2x² = (y')²/2
yy'' - (y')²/2 + y²/2x² = 0

這是屬於http://eqworld.ipmnet.ru/en/solutions/ode/ode0338.pdf類型的微分方程。(差點是屬於http://eqworld.ipmnet.ru/en/solutions/ode/ode0335.pdf類型的微分方程,可是因為(y')²的係數不配合所以不是。)

2009-12-26 05:54:39 補充:
基本上,好像第二題的二階非線性微分方程,若是沒有http://eqworld.ipmnet.ru/index.htm這類的網站照著,隨時會想足十世都未能想到方法求解。

注意:因為第二題是屬於二階非線性微分方程,所以上述的方法我是看http://eqworld.ipmnet.ru/en/solutions/ode/ode-toc3.htm找到的,當然還要題目出得好,因為只要x的次數不一樣就game over!

2010-01-02 08:35:46 補充:

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參考資料:
my wisdom of maths + method from http://eqworld.ipmnet.ru/en/solutions/ode/ode0338.pdf

2010-01-05 17:47:30 補充:
至於第1題這題近乎exact的de,應該是與Abel
equation of the second kind有關,有篇這樣好的文章(http://www.ae.uiuc.edu/lndvl/Publications/2002_IJND.pdf),我會盡力嘗試。就算最後顯示出它是無法solve analytically也是值得的。

2010-01-08 08:26:27 補充:
+ http://www.ae.uiuc.edu/lndvl/Publications/2002_IJND.pdf#page=04 + http://eqworld.ipmnet.ru/en/solutions/ode/ode0125.pdf + http://eqworld.ipmnet.ru/en/solutions/ode/ode0124.pdf + http://www.ae.uiuc.edu/lndvl/Publications/2002_IJND.pdf#page=02

2012-06-02 09:16:25 補充:
喜訊:有人終於在2011年8月找到所有Abel equation of the first kind的解析解,詳見http://www.hindawi.com/journals/ijmms/2011/387429/#sec2。
2009-12-26 9:49 am
題目無誤嗎?
還是題目要求圖解或討論解的性質?

2009-12-26 01:52:36 補充:
Q1:若是 y'=(x^2 y-4x+8xy^3)/(x^3+12x^2 y^2), 則可解
Q2:若是 x^2yy"+y^2=x^2y'^2, 則可解
2009-12-26 9:10 am
初步算了一下~~好像沒那麼容易


收錄日期: 2021-05-04 00:49:02
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