= Solution
Since $\lambda_1=-4/5$, the <kernel of a linear map> $L+\tfrac45I$ is the span of $\psi(x)=\cos(3x/2)$. The self-adjoint <Fredholm alternative> requires $\langle f,\psi\rangle=0$ for solvability. If it holds, an arbitrary multiple of $\psi$ can be added, so a solution is not unique.
For $f=\cos x$,
$$
\int_0^\pi\cos x\cos(3x/2)\,dx=\frac65\ne0,
$$
so \b[case (A) has no solution]. For $f=\sin x\sin(3x/2)$,
$$
\int_0^\pi f(x)\psi(x)\,dx=\frac12\int_0^\pi\sin x\sin3x\,dx=0
$$
by sine-function <orthogonality>. Thus \b[case (B) has solutions, and they are nonunique].
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