3. Derivation of resonance response
Now, add on an external force \(F_{ext}=F\cos(\Omega t)\). We have assumed that this external force is a periodic driving force, with angular frequency \(\Omega\) that is generically different from \(\omega\). Note that \(F\) is a constant.
Initial conditions: Assume that at \(t=0\), we have \(v=0\) and \(x=A>0\).
3.1 Write down the equation of motion of the mass in terms of \(x\) and its time derivatives. Arrange it in a form \(\ddot{x} + 2 \gamma \dot{x} + \omega^2 x = P(t)\), where \(P\left(t\right)\) is some function of time only (and not a function of \(x\)).