Continued discussing the damped harmonic oscillator, then introduced a driving force, making the differential equation inhomogeneous.
First, considered the overdamped oscillator in the limit when 
. We Taylor expanded the general solution about 
, keeping the first two independent solutions (since we continue
to solve a second order equation, with two independent solutions). The
other terms must vanish in the 
limit. We
found two independent pieces survived the limit: the two solutions
and 
claimed for the critically damped ( 
) case. Again we return
to equilibrium without oscillation.
We then considered the late time behavior in each physical case:
1. Underdamped. Remains sinusoidal, with exponentially declining
amplitude 
.
2. Overdamped. Dominated by the 
solution (unless initial conditions set a to zero). This amplitude
decays more slowly than both the underdamped case and the critically
damped case below.
2. Critically damped. Dominated by 
solution, with
rate of change shown to be dominated by term 
. This decays, at late times, much faster than either 1) or 2)
above. Generally, a critically damped spring will return the fastest,
when displaced from equilibrium.
We then considered adding a driving force to the above system. For
simplicity, we first took the force to be 
.
We searched for a particular solution of form 
,
and found one algebraically, by substitution into the differential
equation. This had the form 
, where the expressions
we obtained for a and 
were quite complicated. Thus the
particular solution to this problem was 
.
-- KB