Consider the differential equation $ -y''(x)+x^{4}y(x)-E\_{n}y(x)=0 $ with the boundary conditions $ y(0)=0=y(\infty).$


How can I use the shooting method or other numerical methods to solve this equation? My only idea is to set $ R=10000 $ for example and solve $ y(0)=0=y(R) $,


Of course I also could make the substitution $u= \frac{x}{x-1}$ so the new boundary conditions could become $ y(0)=0=y(1) $ but now the differential equation becomes singular at the point $ u=1$.



How can I solve a boundary value numerically on an infinite interval?
You're facing a singular Sturm-Liouville eigenvalue problem. One way you may be able to use the shooting method is to get an asymptotic expansion of your solution at the $x=\infty$ (Frobenius or WKB methods should work). Using the asymptotics to evaluate $y(x)$ and $y'(x)$ at some finite $x$ and then shoot from there to $x=0$ to find the eigenvalue.


On the other hand, there exists a fairly well developed numerical library, based also on a sound mathematical analysis of the Sturm-Liouville problem, that is already available. It should be able to handle your problem directly. Look up the [SLEIGN2](http://www.math.niu.edu/SL2/) package and its documentation.


