Homogenous first-order differential equation

A first-order ODE in the form

𝑀 (𝑥, 𝑦) 𝑑 𝑥 + 𝑁 (𝑥, 𝑦) 𝑑 𝑦 = 0

is homogenous of order $𝑛$ if the functions $𝑀 (𝑥, 𝑦)$ and $𝑁 (𝑥, 𝑦)$ are both homogenous of order 𝑛;

𝑀 (𝑠 𝑥, 𝑠 𝑦) = 𝑠 𝑛 𝑀 (𝑥, 𝑦) 𝑁 (𝑠 𝑥, 𝑠 𝑦) = 𝑠 𝑛 𝑁 (𝑥, 𝑦)

Such an ODE may be converted to a Separable differential equation using either the substitution

𝑦 = 𝑢 𝑥 ⟹ 𝑑 𝑦 = 𝑥 𝑑 𝑢 + 𝑢 𝑑 𝑥 𝑥 = 𝑣 𝑦 ⟹ 𝑑 𝑣 = 𝑦 𝑑 𝑣 + 𝑣 𝑑 𝑦

The former is advantageous if $𝑁 (𝑥, 𝑦)$ is easier to integrate.

Motivation

Because of the characteristic property of homogenous functions, the ratio of the functions can be expressed as a function of a single variable $𝑢 = 𝑥 / 𝑦$ . Let $𝑠 = 𝑥 - 1$ . Then,

𝑀 (𝑥, 𝑦) 𝑁 (𝑥, 𝑦) = 𝑀 (𝑠 𝑥, 𝑠 𝑦) 𝑁 (𝑠 𝑥, 𝑠 𝑦) = 𝑀 (1, 𝑥 / 𝑦) 𝑁 (1, 𝑥 / 𝑦) = 𝑀 (1, 𝑢) 𝑁 (1, 𝑢)

Practice problems

https://www.mathsisfun.com/calculus/differential-equations-homogeneous.html

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