Compute the eigenvalues and eigenfunctions of $L = −\dfrac{d^2}{dx^2} $.
$$D(L) = \{ f ∈ C^2([0, 1], C)\mid f'(0) = f'(1) = 0\}$$
and
$$L = \frac{1}{r(x)} \left( -\frac{d}{dx} p(x)\frac{d}{dx} + q(x) \right).$$
I'm trying to do the following. Take $y \in D(L)$, and apply the operator $L[y] = −\dfrac{d^2y}{dx^2} $, then $-y''(x) = \dfrac{1}{r(x)} \left( -\dfrac{d}{dx} p(x)\dfrac{dy}{dx} + q(x) \right)= \dfrac{1}{r(x)} \left( -\dfrac{d}{dx} p(x)y'(x) + q(x) \right)$, therefore
$$-y''(x)r(x) = -(p(x)y'(x))' + q(x) = -p'(x)y'(x)-p(x)y''(x) + q(x).$$
I need to solve the following problem $$y''(x)[r(x) -p(x)]-p'(x)y'(x) + q(x)=0.$$
Now I can not find the eigenvalues and eigenfunctions from here. Can someone help me?