# Definition:Differential Equation

## Contents

## 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.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 1.1$: Introduction - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World