Rectangular-pulse cancellation of an undamped oscillator ($T=2\pi n/{\omega }_0$)

For $x^{\prime \prime }+{\omega }_0^{2}x=F$ during $0<t<T$ and zero force afterwards, with zero initial displacement and velocity, the causal Green function gives $x(t)=F[1-\cos ({\omega }_{0}t)]/{\omega }_0^2$ during the pulse. For $t\ge T$ it gives $x(t)=F[\cos ({\omega }_0(t-T))-\cos ({\omega }_{0}t)]/{\omega }_0^2$. For $F\ne 0$, both outgoing displacement and velocity vanish precisely when $T$ is an integer number of full natural periods. Cancellation requires both initial data for the subsequent homogeneous equation, not only zero displacement at switch-off.