I'm looking for a proof that $Ly = (ky')'$ is self-adjoint. My attempt:
\begin{align} \langle Ly(x), w(x)\rangle &= \int_a^b (k(x)y(x)')'w(x) \,\mathrm{d}x \\ &= \int_a^b (k'(x)y(x)' + k(x)y''(x))w(x) \,\mathrm{d}x \\ &=\quad\color{red}{??} \\ &= \int_a^b y(x)(k'(x)w(x)' + k(x)w''(x)) \,\mathrm{d}x \\ &= \langle y(x), Lw(x)\rangle \end{align}
Except I'm not sure how to fill in the center part.