Ordinary differential equations Definition

Homogeneous first-order equation

Builds on Ordinary differential equation Separable 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.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Separable equation definition
Homogeneous first-order equation definition

Tag	`0K4ZWTFSV5`
URI	`m2://0K4ZWTFSV5`
Kind	definition
Writing	W0 stub
MSC	34A05
Defines	Homogeneous first-order equation
Uses	Ordinary differential equation, Separable equation

Homogeneous first-order equation

Definition

Remarks

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