= Solution
For the real <Sturm-Liouville operator> $\mathcal Ly=-(py')'+qy$, two integrations by parts give the <Lagrange identity for a Sturm-Liouville operator>:
$$
\int_0^1(u\mathcal Lv-v\mathcal Lu)\,dx=[p(vu'-uv')]_0^1.
$$
Set $u=y_n,v=\widetilde y$. Both vanish at the endpoints, so the boundary term is zero. Since $\mathcal Ly_n=\lambda_nwy_n$,
$$
\int_0^1y_n\mathcal L\widetilde y\,dx=\int_0^1\widetilde y\mathcal Ly_n\,dx=\lambda_n\int_0^1wy_n\widetilde y\,dx.
$$
Consequently
$$
\boxed{\int_0^1y_nF_n\,dx=\int_0^1y_n(\mathcal L\widetilde y-\lambda_nw\widetilde y)\,dx=0.}
$$
This is <self-adjointness> with the homogeneous <Dirichlet boundary conditions>, and expresses the <Fredholm solvability condition> at the <eigenvalue> $\lambda_n$. In this unweighted integral $F_n$ already contains its factor of $w$; adding another weight would be incorrect.
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