Past exam of the mathematical methods I course of the University of Oxford / 2017 resit / 7 / d / Solution

Divide the external force by $m$. Away from resonance a particular solution has the forcing frequency, so the general forced harmonic oscillator solution isThe two homogeneous terms cannot be discarded in an undamped system: they do not decay. If the same initial conditions are imposed, then $A=x_0-C\cos \varphi$, $B=-C(\omega /{\omega }_0)\sin \varphi$, where $C=F_0/[m({\omega }_0^2-{\omega }^2)]$.

For $\omega \gg {\omega }_0$, the forced displacement approaches $-F_0\cos (\omega t-\varphi )/(m{\omega }^2)$: 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 isIts amplitude grows linearly because the external force supplies energy coherently. For $x(0)=x_0$, $\dot{x}(0)=0$, choose $A=x_0$ and $B=F_0\sin \varphi /(2m{\omega }_0^2)$. Physical damping or nonlinear response ultimately limits the ideal resonant growth.