Solution
= Solution
For $t>1$, every forcing point has $\tau<t$. Hence the <Green's function> gives
$$
x(t)=-\frac14e^{-2t}\int_0^1\tau e^{2\tau}\,d\tau.
$$
Since $\int\tau e^{2\tau}d\tau=(\tau/2-1/4)e^{2\tau}$, the remaining <integral> is $(e^2+1)/4$. Therefore
$$
\boxed{x(t)=-\frac{e^2+1}{16}e^{-2t},\qquad t>1.}
$$
This decays at positive infinity.