Differential equations MOC

Homogenous first-order differential equation

A first-order ODE in the form

𝑀(𝑥,𝑦)𝑑𝑥+𝑁(𝑥,𝑦)𝑑𝑦=0

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

𝑀(𝑠𝑥,𝑠𝑦)=𝑠𝑛𝑀(𝑥,𝑦)𝑁(𝑠𝑥,𝑠𝑦)=𝑠𝑛𝑁(𝑥,𝑦)

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

𝑦=𝑢𝑥⟹𝑑𝑦=𝑥𝑑𝑢+𝑢𝑑𝑥𝑥=𝑣𝑦⟹𝑑𝑣=𝑦𝑑𝑣+𝑣𝑑𝑦

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(𝑠𝑥,𝑠𝑦)𝑁(𝑠𝑥,𝑠𝑦)=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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