Leading nonlinear eigenvalue shift in a Sturm-Liouville problem

For $L(y;\lambda )=y^{m+1}$ near a normalized simple null mode $y_0$, put $y=\epsilon y_0+O({\epsilon }^2)$ and $\lambda -{\lambda }_0={\epsilon }^m\mu$. Orthogonality to $y_0$ yields