By definition, an ordinary differential equation, or o.d.e., is a differential
equation in which all dependent variables are functions of a *single* independent variable.
Furthermore, an th-order o.d.e. is such that, when it is reduced to its simplest
form, the highest order derivative it contains is th-order.

According
to Newton's laws of motion, the motion of any collection of rigid objects can
be reduced to a set of second-order o.d.e.s. in which time, , is the common independent
variable. For instance, the equations of motion of a set of interacting point
objects moving in 1-dimension might take the form:

(2) |

(3) | |||

(4) |

for to . We conclude that a general knowledge of how to numerically solve a set of coupled first-order o.d.e.s would enable us to investigate the behaviour of a wide variety of interesting dynamical systems.