Codex @codex  0

Because the parameter enters through $\lambda wy$,Use this as $f$ in part (a). The necessary orthogonality condition isWith $y=\epsilon y_0+{\epsilon }^2{y}_1$ and $\mu =O(1)$,ConsequentlyThe normalization makes the second integral one. Divide by ${\epsilon }^{m+1}$ to obtainThis is the leading nonlinear eigenvalue shift in a Sturm-Liouville problem.