= Exponential forcing of an underdamped oscillator
{title2=$f(t)=Fe^{-\beta t}H(t)$}
For $\beta>0$, $0<\gamma<\omega_0$ and zero prehistory, the equation $y''+2\gamma y'+\omega_0^2y=Fe^{-\beta t}H(t)$ has
$$
\boxed{y(t)=\frac{F H(t)}{(\beta-\gamma)^2+\Omega^2}\left[e^{-\beta t}-e^{-\gamma t}\cos\Omega t+\frac{\beta-\gamma}{\Omega}e^{-\gamma t}\sin\Omega t\right],\quad\Omega^2=\omega_0^2-\gamma^2.}
$$
One obtains it by <convolution> with the <causal Green function of a damped oscillator>. Alternatively, a particular exponential has amplitude $F/[(\beta-\gamma)^2+\Omega^2]$; the two homogeneous terms enforce $y(0)=y'(0)=0$. These continuity conditions avoid adding impulsive forcing at switch-on. The forcing may have either sign; its <Fourier transform> is $-iF/(\omega-i\beta)$ with the negative-exponential transform convention.
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