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.
Differential equation leading behavior
3
$\begingroup$
ordinary-differential-equations
asymptotics
1 Answers
2
Use the WKP approximation method. See the example which your problem is a special case of it.