= Solution
Write $v=Ly$ as
$$
v(x)=\cos x\int_x^\pi\sin t\,y(t)\,dt+\sin x\int_0^x\cos t\,y(t)\,dt.
$$
Differentiating twice, the moving-limit terms cancel at first order and yield $\boxed{v''+v=y}$ at second order. Also $v'(0)=0$ and $v(\pi)=0$. Thus this <integral operator> is the inverse of $D^2+1$ with mixed <Neumann boundary condition> and <Dirichlet boundary condition>.
For $\kappa_n=n+1/2$, $y_n(x)=\cos(\kappa_nx)$ satisfies those conditions and $(D^2+1)y_n=(1-\kappa_n^2)y_n$. The corresponding mixed homogeneous equation has no nonzero solution, so uniqueness gives
$$
\boxed{Ly_n=\lambda_ny_n,\qquad\lambda_n=\frac1{1-\kappa_n^2}=-\frac4{4n^2+4n-3}.}
$$
This verifies the asserted <eigenvalues> and <eigenfunctions> for every $n\geq0$.
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