# Definition:Differential Equation/Order

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## Definition

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

## Also known as

Some sources refer to the **order** of a differential equation as its **dimension**.

## Examples

### First Order

The following ordinary differential equations are of the $1$st order:

- $\dfrac {\d y} {\d x} + y = 0$

- $y' = e^x$

- $x y' = 2 y$

### Second Order

The following ordinary differential equations are of the $2$nd order:

- $\dfrac {\d^2 y} {\d x^2} = \dfrac 1 {1 - x^2}$

- $\map {f'} x = \map {f''} x$

- $y'' + \paren {3 y'}^3 + 2 x = 7$

### Third Order

The following ordinary differential equation is of the $3$rd order:

- $\paren {y'''}^2 + \paren {y''}^4 + y' = x$

### Fourth Order

The following ordinary differential equation is of the $4$th order:

- $x y^{\paren 4} + 2 y'' + \paren {x y'}^5 = x^3$

## Warning

Consider the ordinary differential equation:

- $y'' - y'' + y' - y = 0$

At first glance it looks as though it is of the $2$nd order.

But after some (fairly obvious) simplification, it is seen that it can be written:

- $y' - y = 0$

which is of the $1$st order.

## Sources

- 1926: E.L. Ince:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.1$ Definitions - 1952: H.T.H. Piaggio:
*An Elementary Treatise on Differential Equations and their Applications*(revised ed.) ... (previous) ... (next): Chapter $\text I$: Introduction and Definitions. Elimination. Graphical Representation: $3$. Definitions - 1956: E.L. Ince:
*Integration of Ordinary Differential Equations*(7th ed.) ... (previous) ... (next): Chapter $\text {I}$: Equations of the First Order and Degree: $1$. Definitions - 1960: D.R. Bland:
*Vibrating Strings*... (previous) ... (next): Chapter $1$: Strings of Finite Length: $1.1$ Introduction - 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: Definition $3.2$ - 1968: G. Stephenson:
*An Introduction to Partial Differential Equations for Science Students*... (previous) ... (next): Chapter $1$ Basic Concepts: $1.1$ Introduction - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 1$: Introduction - 1977: A.J.M. Spencer:
*Engineering Mathematics: Volume $\text { I }$*... (previous) ... (next): Chapter $1$ Ordinary Differential Equations: $1.1$ Introduction: Classification of Differential Equations - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**differential equation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**differential equation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**order**:**2.**(of a differential equation) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**order**(of a differential equation)