here,k,a,b,c have definite values. x\in (0,+\infty). The boundary conditions are: 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 y_is. But the result is wrong.

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