Degree in homogeneous function in differential equations

Question

How do we say $n$ to be degree of an equation,

We have $F(kx,ky)=k^{n} F(x,y)$ then we say n is the degree of the equation but we generally consider the degree to be the highest power of a variable in a polynomial but here the $k$ is an arbitrary constant so how do we determine the degree of the equation with k?

yes you are right I thought it would be considered differential equations as it was used to identify equations with degree 0 but on its own its an homogenous equation only — Razz, Aug 16 '23 at 18:35

score 0 · Accepted Answer · answered Aug 16 '23 at 18:08

0

You have to take into account the fact that you are dealing with two distinct concepts of degree here. You have defined the degree of a polynomial, but the concept of degree of a homogeneous function is a different concept.

Take, for instance, $F(x,y)=x^3+x^2y+xy^2+y^3$. Then, if $k$ is any number, you have\begin{align}F(kx,ky)&=k^3x^3+k^3x^2y+k^3xy^2+k^3y^3\\&=k^3\left(x^3+x^2y+xy^2+y^3\right)\\&=k^3F(x,y),\end{align}and therefore $F$ is a homogeneous function of degree $3$.

answered Aug 16 '23 at 18:08

José Carlos Santos

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do we define the degree of a homogeneous function to be the power of the constant or is there something more to it – Razz Aug 16 '23 at 18:36
There is nothing more to it: the degree is just the number $n$ such that you always have $F(kx,ky)=k^nF(x,y)$. – José Carlos Santos Aug 16 '23 at 18:48
ok thank you for the information – Razz Aug 16 '23 at 19:05
I'm glad I could help. – José Carlos Santos Aug 16 '23 at 19:06

Degree in homogeneous function in differential equations

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