Solution
= Solution
For $t<0$, all forcing points satisfy $\tau>t$, so the <Green's function> formula becomes
$$
x(t)=-\frac14e^{2t}\int_0^1\tau e^{-2\tau}\,d\tau.
$$
An <integration by parts> gives $\int\tau e^{-2\tau}d\tau=-(\tau/2+1/4)e^{-2\tau}$. Consequently
$$
\boxed{x(t)=-\frac{1-3e^{-2}}{16}e^{2t},\qquad t<0.}
$$
The exponential has the required decay as $t\to-\infty$.