= Solution
{c}
At $l=\pi$, the <Fredholm solvability condition> is
$$
0=\int_0^\pi(a\sin x+b\sin2x)\sin x\,dx=\frac{a\pi}{2},
$$
by <orthogonality> of the <Fourier sine basis>. Thus a non-unique solution exists exactly when $a=0$, with any real $b$. Since $(d^2/dx^2+1)\sin2x=-3\sin2x$, the complete family is
$$
\boxed{a=0:\quad y(x)=C\sin x-\frac b3\sin2x,\quad C\in\mathbb R.}
$$
If $a\ne0$ there is no solution, for any $b$. In particular, when $a=b=0$, the family reduces to the <kernel> $C\sin x$; it is still non-unique. There is no choice of $a,b$ producing a unique solution for this length.
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