Many problems in mechanics gives rise to second order differential equations as a consequence of Newton's second law of motion,
The acceleration a is actually the second differential of position with respect to time,
so any problem involing acceleration results in a second order differential equation.
Different types of forces often occur. The force may depend
on position,
as when a particle is moving up or down a slope. If as is often the case
which represents a restoring force towards an equilibrium position (and this is approximately true for small oscillations for many systems).on the speed of the particle, as when the particle experiences a drag force. Often the drag force,
is proportional to the speed (and always in the opposite direction to the motion of the particle), so
(since
).the force may be a drive force
This is an externally applied force, and represents an external source of energy. This may result in the system vibrating wildly, if the frequency of the forcing term is close to the natural frequency of the system.
Substitute these into
to get
Now use
andto get
This is often written as
