→ BLUEBL00D : 通解可再化簡為 y = x^6/4 + cx^3 + c^2 01/05 13:58
※ 引述《xy210742 (Sam)》之銘言：
: (y')^2=9*(x^4)*y
: 請問這題應如何假設變數
: 才能使此題變成線性微分方程
: 謝謝
let z = y' = (D_x) y
=> y = z^2 / (9x^4)
D_x
===> z = [ 2zz'/(9x^4) ] + [ -4z^2/(9x^5) ]
=> z * [ z' - 2z/x - 9x^4/2 ] = 0
===================================================
<a> 通解
z' - 2z/x = 9x^4/2
=> z = 3x^5/2 + cx^2
=> y = z^2 / (9x^4) = x^6/4 + cx^3/3 + c^2/9
===================================================
<b> 異解
z=0 => y = 0^2 / (9x^4) =0
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