---
tag: 0P97V20RW7
kind: definition
name: Ordinary differential equation
aliases: [ode]
msc: [34A99]
defines: 0NT64JV96A
uses: [003R0V5M7V, 0EPAT51DXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Ordinary_differential_equation
    locator: Definition
---

## Definition

Let $n \ge 1$ be an integer, let $U$ be a subset of $\mathbb{R}^{n+2}$, and let $F : U \to \mathbb{R}$ be a function.
An ordinary differential equation of [[order|0XK0G19SK8]] $n$ is an equation
$$F\left(x, y, y', \dots, y^{(n)}\right) = 0$$
for an unknown real function $y$ of one real variable $x$. Here $y^{(k)}$ is the $k$-th derivative of $y$.
The equation must depend on $y^{(n)}$.
If $y$ takes values in $\mathbb{R}^m$ and $F$ takes values in $\mathbb{R}^m$, the equation is a system of $m$ ordinary differential equations.

## Remarks

The word ordinary means that the unknown function depends on one variable only. An equation with partial derivatives of a function of several variables is a partial differential equation.
The equation is in [[explicit form|08WWTN738R]] when it is solved for the highest derivative: $y^{(n)} = f\left(x, y, y', \dots, y^{(n-1)}\right)$.
A [[solution|07VX34QXAV]] is a function on an interval that satisfies the equation at each point of the interval.

## Explanations

### L1 human

An ordinary differential equation is a rule that links a function of one variable with its derivatives.
For example, $y'' + y = 0$ links $y$ with its second derivative. The functions $\sin x$ and $\cos x$ satisfy it.