Solution
= Solution
The forcing is $\tfrac12\sin2x$, which lies in the <image> of $L$. Since the <eigenvalue> on $\sin2x$ is $\pi/4$, a particular solution is $y_p=(2/\pi)\sin2x$. The full family is
$$
\boxed{y(x)=\frac2\pi\sin2x+z(x),\qquad z\perp\cos2x,\quad z\perp\sin2x.}
$$
Thus \b[solutions exist and are not unique]: adding any element of the infinite-dimensional <kernel> leaves $Ly$ unchanged. For example every constant can be added.