For second order DEs that describe forced vibration i've encountered a scenario that confuses me; Damped systems that are not forced, no g(t). Damped systems that are forced.
If there is only a transient solution (exponential decay) then the frequency, amplitude, and phase come from the angle coefficient, and the the C1 and C2.
But if there is a transient and a steady-state solution why are those things i've mentioned above only coming from the steady-state? My text reads (about the transient solution) "In many applications, it is of little importance and (depending on the value of gamma) may well be undetectable after only a few seconds". Is this why my book chooses the steady-state to express those quantities?
With unforced systems you only get the transient solution which goes away in presence of damping.
With forced systems the steady state appears to stay alive and active due to the external force. The transient solution in presence of damping decays axponentially and if the force is periodic the steady state borrows its period from the force.