 Solution

For the real Sturm-Liouville operator $\mathcal{L}y=-(p{y}^{\prime }{)}^{\prime }+qy$, two integrations by parts give the Lagrange identity for a Sturm-Liouville operator:Set $u=y_n,v=\tilde{y}$. Both vanish at the endpoints, so the boundary term is zero. Since $\mathcal{L}{y}_n={\lambda }_{n}w{y}_n$,ConsequentlyThis 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.