---
tag: 0Z3YKRQZ48
kind: definition
name: Linear first-order equation
msc: [34A30]
defines: 07JH131FVP
uses: [0P97V20RW7, 0Q1D85B8G6]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_differential_equation
    locator: First-order equation with variable coefficients
---

## Definition

Let $I$ be an open [[interval|0Q1D85B8G6]] and let $p, q : I \to \mathbb{R}$ be continuous functions.
The linear first-order equation with coefficient $p$ and right side $q$ is the equation
$$y' + p(x)\, y = q(x)$$
for an unknown differentiable function $y : I \to \mathbb{R}$.
It is homogeneous if $q = 0$ on $I$, and inhomogeneous otherwise.

## Remarks

The [[solution formula|06B9W45XV0]] gives each solution explicitly, with an [[integrating factor|01THH3Y3YP]].
The homogeneous equation $y' + p(x) y = 0$ has the solutions $y = C e^{-P(x)}$, where $P$ is an antiderivative of $p$ and $C$ is a constant.
Many authors write the equation as $y' = a(x) y + b(x)$, with $a = -p$ and $b = q$.

## Explanations

### L1 human

The equation is linear because $y$ and $y'$ appear only to the first power, with coefficients that depend on $x$.
There is no product such as $y\, y'$ and no function such as $\sin y$.
This structure is the reason why there is a formula for all the solutions.