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Show that the solution of $x^{3}y''=y$ whose leading behavior as $x\rightarrow0$ is $e^{-2x^{-1/2}}$ is actually given by $x^{3/4}e^{-2x^{-1/2}}$. Do this by writing $y=e^{S(x)}$ and finding the subleading part of S.

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Use the WKP approximation method. See the example which your problem is a special case of it.