Definition:Differential Equation

Definition
A differential equation is a mathematical equation for an unknown function of one or several variables relating:
 * $(1): \quad$ The values of the function itself
 * $(2): \quad$ Its derivatives of various orders.

Ordinary and Partial Differential Equations
There are two types of differential equation:

Linear and Non-Linear
Differential equations can also be classified as to whether they are linear or non-linear.

Linear
A linear differential equation is one where any dependent variables and their derivatives appear to the first power.

Neither are products of dependent variables allowed.

Non-Linear
A non-linear differential equation is one which is not linear.

Autonomous System
A differential equation or system of differential equations is called autonomous if none of the derivatives depend on the independent variable.

The $n$th order autonomous differential equation takes the form:
 * $y^{\left({n}\right)} = f \left({y, y', y'', \dots, y^{\left({n-1}\right)}}\right)$

Explicit System
A differential equation or system of differential equations is called explicit if it can be written in the form:


 * $y^{\left({n}\right)} = f \left({x,y, y', y'', \dots, y^{\left({n-1}\right)}}\right)$

An ODE that is not explicit is implicit.

In practice the vast majority of ODEs are explicit; since such systems can be reduced to a first order problem, the theory of ODEs is concerned mainly with first order problems.