Ordinary differential equation

In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable.

for the motion of a particle of mass m. In general, the force f depends upon the position of the particle x, and thus the unknown variable x appears on both sides of the differential equation, as is indicated in the notation f(x).

Ordinary differential equations are to be distinguished from partial differential equations where there are several independent variables involving partial derivatives.

Ordinary differential equations arise in many different contexts including geometry, mechanics, astronomy and population modelling. Many famous mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoullis, Riccati, Clairaut, d'Alembert and Euler.

Much study has been devoted to the solution of ordinary differential equations. In the case where the equation is linear, it can be solved by analytical methods. Unfortunately, most of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. Approximate solutions are arrived at using computer approximations (see numerical ordinary differential equations).