Problem:

$\displaystyle (\frac{e^{-k^2x^2}}{x}y')'-\frac{a*e^{-k^2x^2}*(b+cx^2)^2}{x}y+\frac{e^{-k^2x^2}*\lambda}{x}y=0$

here,k,a,b,c have definite values. $\displaystyle x\in (0,+\infty)$. The boundary conditions are: $\displaystyle y(0)=0,y'(\infty)=0$. How can I calculate the lowwest eignvalue and the corresponding eignfunction? I have already know that the eignvalue is around 2.

My methods:

I have used SLEIGN2 to get the eignvalue. In order to get the corresponding eignfunction, I take x from 0 to 10 (it's a good approximation), and discretise the equation, and then Gauss elimination. However, because the rightside of the linear functions are all zero, I have no idea but let y_N be a small number, and get all other $\displaystyle y_i$s. But the result is wrong.

Who can give me a suggestion on how can I calculate the eignfunction? Thanks very much!