---
tag: 0K4ZWTFSV5
kind: definition
name: Homogeneous first-order equation
msc: [34A05]
defines: 0RN88KZBJ3
uses: [0P97V20RW7, 0V4E623THY]
sources:
  - url: https://en.wikipedia.org/wiki/Homogeneous_differential_equation
    locator: Homogeneous first-order differential equations
---

## Definition

Let $J$ be an open interval and let $F : J \to \mathbb{R}$ be a continuous function.
A homogeneous first-order equation is an [[ordinary differential equation|0P97V20RW7]] of the form
$$y' = F\left(\frac{y}{x}\right)$$
for an unknown function $y$ of $x$. Its domain is an interval that does not contain $0$, with $y(x)/x \in J$.

## Remarks

An equivalent form is $M(x, y)\, dx + N(x, y)\, dy = 0$, where $M$ and $N$ are homogeneous functions of the same degree $d$: $M(t x, t y) = t^d M(x, y)$ and $N(t x, t y) = t^d N(x, y)$ for $t \ne 0$.
The substitution $y = x v$ gives $x v' = F(v) - v$. This is a [[separable equation|0V4E623THY]] for $v$.
Example: $y' = (x + y)/x = 1 + y/x$ gives $x v' = 1$, so $v = \ln|x| + C$ and $y = x \ln|x| + C x$.
This use of the word homogeneous is different from its use for [[homogeneous linear equations|0EQZSAB9EC]].

## Explanations

### L1 human

In a homogeneous equation, the slope $y'$ depends only on the ratio $y/x$.
So the slope field is the same along each line through the origin.
The new unknown $v = y/x$ captures this, and the equation becomes separable.