Consider a body of mass attached to a spring with spring constant
. The differential equation , where is the vertical
displacement, can be used to model the dynamics of this system. However, such a differential
equation implies that, once the body starts moving, it will oscillate forever. A better
model is one that also considers the effects of frictional forces. If we add a frictional
force that is proportional to the velocity of the body, we get the following differential
equation where
.

Letting and , there
are three possible behaviours of the solution to this equation which we can define in terms
of and as follows: over damped when
; critically damped when ;
and underdamped or oscillatory when . Which of the
following solutions can be used to describe one of these behaviours:

- , where
and
- where