Ordinary differential equations

By ODEs we mean equations involving derivatives with respect to a single variable, usually time. Given that y is a function of x and that y', y'', ...,y⁽ⁿ⁾ denote the derivatives: is the first derivative with respect to x, and y⁽ⁿ⁾ is the nth derivative with respect to x. An ordinary differential equation (ODE) is an equation involving x, y, y', y'', ...,y⁽ⁿ⁾.

ODEs occur very commonly in physics, economics, chemistry, etc:

= 0 - The decay equation.
= 0 - The growth equation.
= 0 - The oscillation equation.

The order of a differential equation is the order n of the highest derivative that appears.

When a differential equation of order n has the form

F(x, y', y'', y⁽ⁿ⁾) = 0

it is called an implicit differential equation whereas the form

F(x, y', y'', y^(n-1)) = y⁽ⁿ⁾

is called an explicit differential equation.

An ODE of order n is said to be linear if it is of the form

A linear ODE where Q(x) = 0 is said to be homogeneous. Confusingly, an ODE of the form

is also sometimes called "homogeneous." A differential equation not depending on x is called autonomous.