= Solution
{c}
At $a=\pi$, the nonzero <homogeneous solution> $e^{-x}\sin x$ satisfies both zero <boundary conditions>. Thus the <boundary value problem> has a nontrivial <kernel> and cannot have an inverse for arbitrary forcing.
More directly, the <adjoint operator> has the zero-endpoint <homogeneous solution> $w(x)=e^x\sin x$. If an ordinary <Green's function> existed, the <integration by parts> identity would imply
$$
0=\int_0^\pi w(x)\mathcal L_xg(x,\xi)\,dx
=\int_0^\pi w(x)\delta(x-\xi)\,dx=w(\xi).
$$
But $w(\xi)\ne0$ for $0<\xi<\pi$. This contradiction rules out the <Green's function> for the unmodified operator. It is the <Fredholm alternative> obstruction.
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