Sturm-Liouville expansion with no lambda?

[renegade05]

How would I solve this :

$\displaystyle -(xu')'=\frac{1}{x} \ln{x} \enspace 1<x<e$

$\displaystyle u(1)=0 \enspace, u'(e)=0$

I want to expand this with:

$\displaystyle u(x) = \sum_{n=1} ^{\infty} b_n X_n(x)$

Where $\displaystyle X_n(x)$ are the eigenfunctions of a S-L problem.

$\displaystyle b_n$ are some constants.

I want to express the solution as a single integral, from x = 1 to x = e.

Can someone please show me how to expand this? How can I find the eigenfunctions and the b_n. I know it is easy to solve with different methods - but I would like to do it this way to get a better grasp of similar problems. Thanks!

The lack of a lambda is really throwing me off - no idea how to do this.

 

[Ishuda]

How would I solve this :

$\displaystyle -(xu')'=\frac{1}{x} \ln{x} \enspace 1<x<e$

$\displaystyle u(1)=0 \enspace, u'(e)=0$

I want to expand this with:

$\displaystyle u(x) = \sum_{n=1} ^{\infty} b_n X_n(x)$

Where $\displaystyle X_n(x)$ are the eigenfunctions of a S-L problem.

$\displaystyle b_n$ are some constants.

I want to express the solution as a single integral, from x = 1 to x = e.

Can someone please show me how to expand this? How can I find the eigenfunctions and the b_n. I know it is easy to solve with different methods - but I would like to do it this way to get a better grasp of similar problems. Thanks!

The lack of a lambda is really throwing me off - no idea how to do this.

Strictly speaking this is not a S-L problem, it is just a normal second order ODE.

 

[renegade05]

Strictly speaking this is not a S-L problem, it is just a normal second order ODE.

Ya, asked to find a suitable S-L problem so I can use the eigenfunctions to solve the PDE.

I managed to solve it

thanks!