Solution
= Solution
The relevant <weighted inner product> is $\langle u,v\rangle_w=\int_0^1w(x)u(x)v(x)\,dx$. The normalized <eigenfunctions> of the <Sturm-Liouville problem> have the <orthonormality> property
$$
\boxed{\int_0^1w(x)y_n(x)y_m(x)\,dx=\delta_{nm}.}
$$
Here $\delta_{nm}$ is the <Kronecker delta>: the integral is one for $n=m$ and zero otherwise. The weight is $w$, rather than $p$ or the unweighted measure.