 Bilateral exponential forcing of an overdamped oscillator

For $a>b>0$ with $a,b\ne 1$, the solution of $(D_t+a)(D_t+b)y=e^{-|t|}$ that decays at both infinities isIts 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 $y,y^{\prime }$ 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.