= Retarded oscillator driven by a two-sided exponential
For $g(t)=\alpha e^{-\alpha|t|}/2$ with $\alpha>0$, convolution gives
$$
Q_R(t)=\frac{\alpha}{2(1+\alpha^2)}
\begin{cases}e^{\alpha t},&t<0,\\e^{-\alpha t}+2\alpha\sin t,&t\ge0.\end{cases}
$$
The coordinate and velocity are continuous at zero. At late times the undamped oscillation has amplitude $\alpha^2/(1+\alpha^2)$; <causality> fixes the <homogeneous solution> only after the no-incoming-oscillation condition is specified.
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