---
tag: 003R0V5M7V
kind: concept
name: Differential equation
msc: [34A99]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_equation
    locator: lead section
same_as:
  wikidata: Q11214
  wikipedia: Differential equation
---

## Summary

A differential equation is an equation that relates an unknown [[function|0EPAT51DXQ]] to one or more of its derivatives.
A solution is a function that makes the equation true at each point of its domain.
If the unknown function depends on one variable, the equation is an [[ordinary differential equation|0NT64JV96A]]. If it depends on several variables and the equation has partial derivatives, it is a partial differential equation.

## Remarks

The [[order|0XK0G19SK8]] of a differential equation is the highest order of a derivative in it.
The equation $y' = y$ is a differential equation of order $1$. Its solutions are the functions $y(x) = C e^{x}$ with a constant $C$.
Differential equations describe how a quantity changes. Many laws of physics, biology and economics have this form.

## Explanations

### L1 human

An ordinary equation asks for a number, but a differential equation asks for a function.
The equation gives a rule for the rate of change of the function.
For example, $y' = y$ says that the function grows at a rate equal to its own value.