= Boundary data in a Sturm-Liouville eigenfunction expansion
{title2=$c_n=\frac{\int u_nf-[p(u_ny'-u_n'y)]_a^b}{(\lambda_n+\mu)N_n}$}
Let $Ay=(py')'+qy$ have real <eigenfunctions> $u_n$ obeying $Au_n=\lambda_nwu_n$ and homogeneous <Dirichlet boundary conditions>, with $N_n=\int_a^bwu_n^2>0$. Suppose $Ay+\mu wy=f$, with possibly nonzero endpoint values. The <Lagrange identity for a Sturm-Liouville operator> gives
$$
\int_a^bu_nf\,dx-(\lambda_n+\mu)\int_a^bwu_ny\,dx=[p(u_ny'-u_n'y)]_a^b.
$$
Division by $(\lambda_n+\mu)N_n$ gives the displayed coefficient whenever the denominator is nonzero. At <resonance in a differential equation>, equality of the forcing projection and the boundary term is the <Fredholm solvability condition>. The <Sturm-Liouville eigenfunction expansion> represents the interior solution in the <weighted inner product> sense; its zero endpoint terms do not reproduce nonzero <boundary traces> by pointwise substitution. A smooth lift of the endpoint data reduces the calculation to homogeneous <Dirichlet boundary conditions>.
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