= Solution
For the zero initial data, the <Green function of an undamped oscillator> is $G_0(x,s)=\sin[k(x-s)]/k$ for $0\leq s\leq x$, and zero for $s>x$. It is continuous at the source and its first derivative jumps by one. The resulting <convolution> is
$$
y(x)=2\int_0^x\sin[k(x-s)]\cos(ks)ds=\boxed{x\sin kx.}
$$
It 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 therefore
$$
G_D(x,s)=-\frac{\sin[\kappa(x_<-a)]\sin[\kappa(b-x_>)]}{\kappa\sin[\kappa(b-a)]},\qquad x_<=\min(x,s),\ x_>=\max(x,s).
$$
Its 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 gives
$$
\boxed{y(x)=\frac{k^2}{2}\int_a^bG_D(x,s)ds=\frac12\left[1-\frac{\cos(k(x-L/2))}{\cos(\pi/12)}\right].}
$$
Equivalently $y=(1-\cos kx)/2+(\sqrt3/2-1)\sin kx$. The negative interior response is consistent with the negative kernel for this short Dirichlet interval.
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