[微積] Euler equation
x^2 y"+阿法xy'+杯塔y=0
Consider x > 0 and let x = e^t
A(d^2y/dt^2)+B(dy/dt)+C=0
Let r1 and r2 be the roots of Ar^2 + Br + C = 0
show thar if r1 and r2 are real and
equal, then y = (c1 + c2t)e^r1t = (c1 + c2 ln x)x^r1
y'=(c1r1+c2+c2r1t)e^r1t
y"=(c1r1^2+c2r1+c2r1+tc2r1^2)e^r1t
c1(Ar1^2+Br1+C)e^r1t+c2(tAr1^2+tBr1+tC+2Ar1t+B)e^r1t
=c1(Ar1^2+Br1+C)e^r1t+c2[t(Ar1^2+Br1+C)+2Ar1t+B]e^r1t
右邊多出2Ar1t+B
沒辦法證明是0
怎麼辦
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因為要改的壞習慣太多了 所以就改天吧
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