= Bilateral exponential forcing of an overdamped oscillator
{title2=$f(t)=e^{-|t|}$}
For $a>b>0$ with $a,b\ne1$, the solution of $(D_t+a)(D_t+b)y=e^{-|t|}$ that decays at both infinities is
$$
y(t)=\begin{cases}
\dfrac{e^t}{(a+1)(b+1)},&t\leq0,\\
\dfrac{e^{-t}}{(a-1)(b-1)}+\dfrac2{a-b}\left(\dfrac{e^{-at}}{a^2-1}-\dfrac{e^{-bt}}{b^2-1}\right),&t\geq0.
\end{cases}
$$
Its <Fourier transform> is $2/[(1+k^2)(a+ik)(b+ik)]$. The lower half-plane contains only the <pole> $-i$, while the upper half-plane contains $i,ia,ib$. Inverting by the <residue theorem> gives the two expressions. Direct substitution verifies the equation separately on each side, and $y,y'$ match at zero, so there is no spurious impulsive source. The negative-time response is nonzero because the forcing already acts before zero, even though the <Green function> is causal. Coinciding rates $a=1$ or $b=1$ require limits with polynomial times exponential terms.
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