Solution
= Solution
The Lagrange identity for this <Sturm-Liouville theory> operator is
$$
y_0L(y;\lambda_0)-yL(y_0;\lambda_0)
=\frac d{dx}\left[p(y_0y'-yy_0')\right].
$$
Integrating from zero to one, the boundary term vanishes because both $y$ and $y_0$ vanish at both endpoints. Since $L(y_0;\lambda_0)=0$ and $L(y;\lambda_0)=f$, it follows that
$$
\int_0^1y_0f\,dx=0.
$$
This is the <solvability condition at a Sturm-Liouville eigenvalue>.