1. ## Sturm-Liouville system

Dear all, how can I transfer the following general Sturm-Liouville equation

$\displaystyle [p(x)y']'+[q(x)+\lambda r(x)]y=0$

into the following special Sturm-Liouville equation

$\displaystyle y''+(\lambda +q(x))y=0?$

Thank you very much.

2. ## Re: Sturm-Liouville system

There are some hints to the problem. It goes as follows.

1. Take $\displaystyle y=u(x)z$, choose appropriate $\displaystyle u(x)$, we can get $\displaystyle \ddot z+A(x)z=0$.

2. Then take $\displaystyle x=f(t)$ for some function $\displaystyle f$, we can see the result.

However, I could not check the second hint.

3. ## Re: Sturm-Liouville system

I'll make one comment. If you do step 1. to eliminate z' then step 2. will put z' back into the ODE. I think what you what to do is steps 1. and 2. together.

Also, I don't think you're going to get the same q(x).

4. ## Re: Sturm-Liouville system

Thank you very much for your help. And I've solved this problem in a very tedious way as you said. Good luck to you.