Retarded oscillator driven by a two-sided exponential

For $g(t)=\alpha e^{-\alpha |t|}/2$ with $\alpha >0$, convolution givesThe 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.