= Solution
Multiply the equation for $y_n$ by $y_m$ and that for $y_m$ by $y_n$, then subtract. Integrating the resulting identity yields
$$
\bigl[p(y_my_n'-y_ny_m')\bigr]_{x_0}^{x_1}
+(\lambda_n-\lambda_m)\int_{x_0}^{x_1}wy_ny_m\,dx=0.
$$
All endpoint terms vanish because both functions satisfy the zero <Dirichlet boundary conditions>. Since the multipliers differ,
$$
\boxed{\int_{x_0}^{x_1}w(x)y_n(x)y_m(x)\,dx=0.}
$$
This is the weighted <orthogonality of Sturm-Liouville eigenfunctions>.
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