= Solution
For the real coefficients of the stated <self-adjoint> <Sturm-Liouville problem>, define the weighted <inner product>
$$
\boxed{\langle u,v\rangle_w=\int_a^b w(x)\overline{u(x)}v(x)\,dx.}
$$
Since $w>0$, a nonzero eigenfunction has positive squared <norm>. Let $L y=-(py')'+qy$. <Integration by parts> and the <Dirichlet boundary conditions> give
$$
\int_a^b\overline u\,Lv\,dx=\int_a^b\bigl(p\overline{u'}v'+q\overline uv\bigr)\,dx=\int_a^b\overline{Lu}\,v\,dx.
$$
Taking $u=v$ to be an <eigenfunction> shows that its <eigenvalue> is real, since $\lambda\int w|u|^2=\int(p|u'|^2+q|u|^2)$ is real. For two eigenfunctions with <eigenvalues> $\lambda,\mu$, the same identity gives
$$
(\mu-\lambda)\langle u,v\rangle_w=0.
$$
Hence \b[distinct eigenvalues imply orthogonality in the weighted inner product]. This is the <orthogonality of Sturm-Liouville eigenfunctions>. The calculation assumes the regularity needed for the stated self-adjoint boundary problem, so all boundary terms and integrals are defined.
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