Dirichlet resonance for the forced sine equation ($y^{\prime \prime }+y=-f,y(0)=y(L)=0$)

A continuous forcing gives the solution with zero initial data $z(x)=-{\int }_0^x\sin (x-s)f(s)ds$. Every solution vanishing at zero is $A\sin x+z(x)$. If $\sin L\ne 0$, the remaining endpoint determines $A$ uniquely, with Green's functionIts first derivative has jump $-1$, the correct sign for $-y^{\prime \prime }-y=f$. If $L=m\pi$ with a positive integer $m$, solvability instead requires ${\int }_0^{L}f(s)\sin sds=0$, and the coefficient $A$ is free. This is the Fredholm alternative with a one-dimensional homogeneous kernel.