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)).
Read more about this topic: Ordinary Differential Equation
Famous quotes containing the word background:
“... every experience in life enriches ones background and should teach valuable lessons.”
—Mary Barnett Gilson (1877?)
“In the true sense ones native land, with its background of tradition, early impressions, reminiscences and other things dear to one, is not enough to make sensitive human beings feel at home.”
—Emma Goldman (18691940)
“They were more than hostile. In the first place, I was a south Georgian and I was looked upon as a fiscal conservative, and the Atlanta newspapers quite erroneously, because they didnt know anything about me or my background here in Plains, decided that I was also a racial conservative.”
—Jimmy Carter (James Earl Carter, Jr.)