Solution

Substituting the particular (i.e. steady-state) solution $x=A\cos(\omega_0t-\phi)$ into the expression $\ddot x+2\gamma\dot x$+\omega_0^2x, we obtain $-2\gamma\omega A\sin(\omega_0t-\phi)$. (Note that the first and third terms cancel out, because the particular solution happens to be the solution to the homogeneous equation $\ddot x+\omega_0^2x=0$.) Thus, the required forcing is $F(t)=-2m\gamma\omega A\sin(\omega_0t-\phi)$.

Conversely, if the forcing has amplitude $F_0$ and $\omega=\omega_0$, then $2m\gamma\omega A=F_0$, and the amplitude of the forced oscillation is $A=F_0/(2m\gamma\omega_0)$. When a constant force $F_0$ is applied instead, the deflection of the oscillator is $F_0/(m\omega_0^2)$. The amplitude of the forced oscillation is thus $\omega_0/(2\gamma)$ times this deflection.