Solvability condition at a Sturm-Liouville eigenvalue

If $L(y_0;{\lambda }_0)=0$ and $y_0,y$ satisfy homogeneous separated boundary conditions, the Lagrange identity givesThus $L(y;{\lambda }_0)=f$ can be solved only if $f$ is orthogonal to the null mode $y_0$.

At a simple eigenvalue of a self-adjoint linear operator, an inhomogeneous equation is solvable only when its forcing is orthogonal to the null eigenfunction. In weakly nonlinear expansions, this condition determines the amplitude equation.

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