※ 引述《a668t (大頭)》之銘言:
: 題目
: x*d^2y/dx^2+(1-2x)*dy/dx+(x-1)y=0
: 要用什麼方法解呢?我解不出來@@ 麻煩了!謝謝
x y'' + (1-2x) y' + (x-1)y = 0
x(y'' - 2y' + y) + ( y' - y ) = 0
x D - (x - 1)
D - 1
2
{ x D + ( x( 0 + (-1) ) - (x - 1) ) D + ( x - 1 ) } y = 0
{ x D - (x-1) }( D - 1 ) y = 0
y' - y = u
xu' - (x-1)u = 0
1
u' + ──(1 - x) u = 0
x
1 x
u = c1 ── e
x
1 x
y' - y = c1 ── e
x
x x 1 -x x
y = c2 e + c1 e ∫ ── e e dx
x
x x
= c2 e + c1 e lnx
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