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Can you help me find the leading asymptotic behaviors about the irregular singular point $x=0$ of $x^4 \frac{d^2y}{dx^2}+ \frac{1}{4}y=0$. I do not know where to start with this

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What do you mean by asymptotic behavior? –  Davide Giraudo Feb 23 '12 at 21:01
I it means is the equation analytic. Im not sure tho. –  steven Feb 23 '12 at 21:10
Mathematica gives the following: inputting "DSolve[x^4 f''[x] + 1/4 f[x] == 0, f[x], x]" gives a general solution of the form "E^(I/(2 x)) x C[1] - I E^(-(I/(2 x))) x C[2]", for some constants C[1] and C[2]. Thus, it seems likely that some variable substitution of your original equation leads a differential equation with constant coefficients. Note that your equation is irregular singular at $x=0$, which explains why the solution has an essential singularity around this point. –  A Walker Feb 23 '12 at 22:20
I have got $y(x) ~ c_{1}8\exp{2/x}+c_{2}8\exp{-2/x}$, is this on the right track for the answer? –  steven Feb 23 '12 at 23:17

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