Homogeneous first-order equation

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 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 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.

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.