 Exponential forcing of an underdamped oscillator

For $\beta >0$, $0<\gamma <{\omega }_0$ and zero prehistory, the equation $y^{\prime \prime }+2\gamma y^{\prime }+{\omega }_0^{2}y=F{e}^{-\beta t}H(t)$ hasOne 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 $y(0)=y^{\prime }(0)=0$. 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.