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For the zero initial data, the Green function of an undamped oscillator is $G_0(x,s)=\sin [k(x-s)]/k$ for $0\le s\le x$, and zero for $s>x$. It is continuous at the source and its first derivative jumps by one. The resulting convolution isIt vanishes together with its first derivative at zero and has the required resonant forcing.

For the two-endpoint problem, write $L=\pi /(6k)$, $a=\min (0,L)$, $b=\max (0,L)$ and $\kappa =|k|$. This covers both signs of the supplied nonzero $k$. Left and right homogeneous solutions are $\sin [\kappa (x-a)]$ and $\sin [\kappa (b-x)]$, with Wronskian $-\kappa \sin [\kappa (b-a)]$. The Dirichlet Green function is thereforeIts endpoint values are zero and its first derivative jumps by one. Here $\kappa (b-a)=\pi /6$, so the denominator does not vanish. Integrating the constant forcing givesEquivalently $y=(1-\cos kx)/2+(\sqrt{3}/2-1)\sin kx$. The negative interior response is consistent with the negative kernel for this short Dirichlet interval.