= Solution
Expand a forcing in the complete <orthonormal> <basis> $\phi_n$. Since $L\phi_n=-\lambda_n\phi_n$, its inverse multiplies the $n$th coefficient by $-1/\lambda_n$. Consequently
$$
\boxed{g(x,t)=-\sum_{n=0}^\infty\frac{\phi_n(x)\phi_n(t)}{\lambda_n}=-2\sum_{n=0}^\infty\frac{\sin(k_nx)\sin(k_nt)}{k_n^2-1}.}
$$
The negative sign follows from the <eigenvalue> convention $Ly=-\lambda y$. This <kernel> satisfies the mixed endpoint conditions in its first argument and $L_xg=\delta(x-t)$.
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