Here is the problem:
$$ x^2y''-xy+\lambda y = 0,\quad y(1)=0,\quad y(L)=0,\quad L>0 $$
I am asked to find the Eigenvalues and Eigenfunction.
I can't figure out how to get a general equation for y. I tried integrating factors but that was a mess, I'm not sure if there's a better way to do this. It might be a Sturm Liouville equation, but I'm not sure to to solve those. Any help would be appreciated.
Here is the solution given:
$ \lambda_n=1+(n\pi/ln(L))^2,\quad y_n(x)=xsin(n\pi ln(x)/ln(L));\quad n=1,2,3... $