= Solution
Divide the <external force> by $m$. Away from resonance a particular solution has the forcing frequency, so the general <forced harmonic oscillator> solution is
$$
\boxed{x(t)=A\cos\omega_0t+B\sin\omega_0t+
\frac{F_0}{m(\omega_0^2-\omega^2)}\cos(\omega t-\phi),\qquad\omega\ne\omega_0.}
$$
The two homogeneous terms cannot be discarded in an undamped system: they do not decay. If the same initial conditions are imposed, then $A=x_0-C\cos\phi$, $B=-C(\omega/\omega_0)\sin\phi$, where $C=F_0/[m(\omega_0^2-\omega^2)]$.
For $\omega\gg\omega_0$, the forced displacement approaches $-F_0\cos(\omega t-\phi)/(m\omega^2)$: its amplitude decreases quadratically and it is in antiphase with the force. This statement concerns the forced component; a freely excited component at $\omega_0$ can remain.
At <resonance> the preceding denominator is zero and the appropriate solution is
$$
\boxed{x(t)=A\cos\omega_0t+B\sin\omega_0t+
\frac{F_0t}{2m\omega_0}\sin(\omega_0t-\phi).}
$$
Its amplitude grows linearly because the external force supplies energy coherently. For $x(0)=x_0$, $\dot x(0)=0$, choose $A=x_0$ and $B=F_0\sin\phi/(2m\omega_0^2)$. Physical damping or nonlinear response ultimately limits the ideal resonant growth.
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