Definition:Differential Equation/Order

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This page is about Order of Differential Equation. For other uses, see 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.


Also see

  • Results about the order of a differential equation can be found here.


Sources

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