Driven damped motion of a mass *m*
is modeled by the 2nd order ODE:

where *x*(*t*) is the distance
from equilibrium, *c* is the damping constant, *k* is the
spring constant and *f* is an external driving force. We divide
by *m* and write:

where 2λ = *c/m*,
ω^{2} = *k/m* and *F* = *f*. The general
solution breaks up into two pieces, the complementary and particular:

where *x _{c}* (the
complementary solution) is

The graph below plots the solution and the decomposition into transient and steady-state terms for the IVP:

Changing the initial velocity changes the transient term, but not the steady-state term. The graph below shows the solution curves that result when the initial conditions are altered. Notice that eventually the graphs looks the same.

When *f* (*t*) =
*F*_{0} sin(ω_{0} *t*),
where ω_{0} = (ω^{2} -
λ^{2})^{½} is the natural frequency of
the system, *resonance* can occur. This is the phenomenon by
which the oscillations grow without bound, potentially ripping the
system apart.

For example, the following IVP models a driven undamped system where the driving frequency equals the natural frequency:

Images generated by sage.