Show that the Sturm-Liouville operator $L$ in $L^2([a,b])$ given by $$L=\frac{1}{r(x)}\left(DpD+q\right)$$ is not symmetric.
I'm assuming $p=p(x)>0$ and $q=q(x)\geq 0$, as described by the problem directly above this problem in the textbook I'm studying from (just so you know). Is that reasonable?
For an operator to be symmetric, it has to satisfy $\langle Lx,y\rangle=\langle x,Ly\rangle$, $\forall x,y\in L^2([a,b])$, correct? This looks the same as the definition for self-adjoint. Are they the same?
I'm not quite sure how to proceed. Any help/hints would be useful.