1. ## Regular S-L problem

Hi, I'm trying to find the eigenfunctions and eigenvalues of

$\displaystyle x''(t)+2x'(t)+\lambda x(t) = 0$

with

$\displaystyle x(0) = x(1) = 0$.

First, should I write in self adjoint form? I found it is

$\displaystyle (e^{2t}x'(t))' + \lambda x(t) = 0$

But I don't know how to continue, all the previous examples ended in a harmonic oscillator type equation. Just a hint will be enough

Thanks.

2. I would just solve the characteristic equation, $\displaystyle r^2+ 2r+ \lambda= 0$.

$\displaystyle r= \frac{-2\pm\sqrt{4- 4\lambda}}{2}= -1\pm\sqrt{1- \lambda}$

Now look at what happens for $\displaystyle \lambda< 1$, $\displaystyle \lambda= 1$, $\displaystyle \lambda> 1$.

Which of those give functions that will satisfy the boundary conditions non-trivially?

3. Hi, thanks for your post!

Well, I found that we have non-trivial solutions only if $\displaystyle \lambda > 1$

and that are given by:

$\displaystyle x_{n}(t)=Ce^{-t}\sin((\sqrt{\lambda_{n}-1})t), \mbox{ where } \lambda_{n} = n^2\pi^2 + 1 \mbox{ and } C \in \mathbb{R}$

4. Yes, that is exactly right!