How to find eigenvalues and eigenfunctions of Sturm-Liouville problem?

Question

I have Sturm-Liouville problem: $x^2u''+xu'+\lambda u=0$

With conditions

$1<x<e^{\pi}$

$u'(1)=0$

$u'({e^\pi})=0$

I wrote this SL problem is self adjoint form:

$\frac{d}{dx}[x \frac{du}{dx}]=-\lambda w(x)u$, where $w(x)=\frac{1}{x}$

and found followings

$x(x'u)'=0$ and $xu'=c_1$ so $u'_1=c1/x$ and by boundary conditions $c_1=0$ but I'm stuck since second equality is $u_1=c_1 lnx+ c_2$ nothing useful here.

Now i need to find its eigenvalues and eigenfunctions. I studied some examples but I did not understand how to find for this case. Can anyone show me a guideline since i have to show eigenfunctions' orthogonality?

Answer 1

As mentioned in comments, take $x = e^z$ and it is very straightforward to show that $x^2u'' = \frac{d^2u}{dz^2} - \frac{du}{dz}$ and $ xu' = \frac{du}{dz}$. Find the corresponding solutions and use boundary conditions to determine Eigen values.

Can you take it from here?