Ordinary Differential Equation - Background

Background

Ordinary differential equations arise in many different contexts throughout mathematics and science (social and natural) one way or another, because when describing changes mathematically, the most accurate way uses differentials and derivatives (related, though not quite the same). Since various differentials, derivatives, and functions become inevitably related to each other via equations, a differential equation is the result, governing dynamical phenomena, evolution and variation. Often, quantities are defined as the rate of change of other quantities (time derivatives), or gradients of quantities, which is how they enter differential equations.

Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), geology (weather modelling), chemistry (reaction rates), biology (infectious diseases, genetic variation), ecology and population modelling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).

Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler.

A simple example is Newton's second law of motion — the relationship between the displacement x and the time t of the object under the force F which leads to the differential equation

for the motion of a particle of constant mass m. In general, F depends on the position x(t) of the particle at time t, and so the unknown function x(t) appears on both sides of the differential equation, as is indicated in the notation F(x(t)).