= Solution
Set $L=d^2/dt^2-4$ and choose $L_tG(t,t')=\delta(t-t')$, with the <Green's function> decaying at both ends. Away from the source, its homogeneous solutions are $e^{\pm2t}$. Decay at each end and continuity at $t=t'$ therefore give $G=C e^{-2|t-t'|}$. Integrating its equation across the source gives a unit jump in its first <derivative>:
$$
\partial_tG(t'^+,t')-\partial_tG(t'^-,t')=-2C-2C=1.
$$
Thus
$$
\boxed{A=-2,\qquad C=-\frac14,\qquad G(t,t')=-\frac14e^{-2|t-t'|}.}
$$
The solution of the <boundary value problem> is the <convolution>
$$
\boxed{x(t)=-\frac14\int_{-\infty}^{\infty}e^{-2|t-\tau|}f(\tau)\,d\tau.}
$$
There is no extra homogeneous term: no nonzero combination of $e^{2t},e^{-2t}$ decays at both infinities. The forcing must of course permit the imposed decay; integrable $f$, including part (b), is sufficient. Literally arbitrary forcing need not do so: constant nonzero forcing admits no solution vanishing at both ends.
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