# Definition:Differential Equation/Order

## 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.