---
tag: 09M2RDB61G
kind: definition
name: Order of a differential equation
msc: [34A99]
defines: 0XK0G19SK8
uses: [0P97V20RW7]
sources:
  - url: https://en.wikipedia.org/wiki/Ordinary_differential_equation
    locator: Definition
---

## Definition

Let $F\left(x, y, y', \dots, y^{(n)}\right) = 0$ be an [[ordinary differential equation|0P97V20RW7]] for an unknown function $y$ of $x$.
Its order is the largest integer $k$ such that the equation depends on the derivative $y^{(k)}$.
For a system of equations, the order is the largest order of a derivative that occurs in the system.

## Remarks

The equation $y' = x y$ has order $1$, and the equation $y'' + \sin y = 0$ has order $2$.
The order does not depend on the powers of the derivatives: $(y')^3 = y$ has order $1$.
An equation of order $n$ can be changed into a first-order system of $n$ equations. See [[reduction to a first-order system|0TH9CPW3VS]].

## Explanations

### L1 human

The order tells you how many times the unknown function is differentiated, at most.
An equation with $y'$ only is of first order. An equation with $y''$, but no higher derivative, is of second order.