# 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.

### Order

The order of a differential equation is defined as being the order of the highest order derivative that is present in the equation.

## Ordinary and Partial Differential Equations

There are two types of differential equation:

### Ordinary Differential Equation

An ordinary differential equation (abbreviated O.D.E. or ODE) is a differential equation which has exactly one independent variable.

All the derivatives occurring in it are therefore ordinary.

The general ODE of order $n$ is:

$f \left({x, y, \dfrac {\mathrm d x} {\mathrm d y}, \dfrac {\mathrm d^2 x} {\mathrm d y^2}, \ldots, \dfrac {\mathrm d^n x} {\mathrm d y^n}}\right) = 0$

or, using the prime notation:

$f \left({x, y, y', y'', \ldots, y^{\left({n}\right)}}\right) = 0$

### Partial Differential Equation

A partial differential equation (abbreviated P.D.E. or PDE) is a differential equation which has more than one independent variable.

The derivatives occurring in it are therefore partial.

## 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 a differential equation where all 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 a differential equation which is not linear.

## Solution

Let $\Phi$ be a differential equation.

Any function $\phi$ which satisfies $\Phi$ is known as a solution of $\Phi$.

Note that, in general, there may be more than one solution to a given differential equation.

On the other hand, there may be none at all.

### Solution Set

The solution set (or the solution) of $\Phi$ is the set of all functions $\phi$ that satisfy $\Phi$.

## Autonomous

A differential equation is autonomous if none of the derivatives depend on the independent variable.

The $n$th order autonomous differential equation takes the form:

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

## System of Differential Equations

A system of differential equations is a set of simultaneous differential equations.

The solutions for each of the differential equations are in general expected to be consistent.

## Explicit and Implicit

### Explicit System

A differential equation is called explicit if it can be written in the form:

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

### Implicit

A differential equation that is not explicit is referred to as implicit.

## Also see

• Results about differential equations can be found here.

## Historical Note

Much of the theory of differential equations was established by Leonhard Paul Euler.

The first existence proof for the solutions of a differential equation was provided by Augustin Louis Cauchy.