# Definition:Differential Equation

## Contents

## Definition

A **differential equation** is a mathematical equation for an unknown function of one or several variables relating:

- The values of the function itself, and
- Its derivatives of various orders.

### Order

The **order** of a differential equation is defined as being that 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 one which has only one independent variable.

All the derivatives occurring in it are therefore ordinary.

The general **ODE** of order $n$ is:

- $\displaystyle f \left({x, y, \frac {dx} {dy}, \frac {d^2x} {dy^2}, \ldots, \frac {d^nx} {dy^n}}\right) = 0$

or, using the prime notation:

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

### Partial Differential Equation

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

The derivatives occurring in it are therefore partial.

### Mixed Differential Equation

A **mixed differential equation** is one in which both ordinary derivatives and partial derivatives occur.

## 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.

## 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 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.

## Sources

- George F. Simmons:
*Differential Equations*(1972): $\S 1, \ \S 2$