According to Wikipedia, there are two definitions for homogeneity of a differential equation. The first is for first order differential equations:
A first-order ordinary differential equation in the form:
${\displaystyle M(x,y)\,dx+N(x,y)\,dy=0}$
is a homogeneous type if both functions $M(x, y)$ and $N(x, y)$ are homogeneous functions of the same degree $n$.[1] That is, multiplying each variable by a parameter ${\displaystyle \lambda }$, we find
${\displaystyle M(\lambda x,\lambda y)=\lambda ^{n}M(x,y)\,}$, and ${\displaystyle N(\lambda x,\lambda y)=\lambda ^{n}N(x,y)\,.} $
Thus, ${\displaystyle {\frac {M(\lambda x,\lambda y)}{N(\lambda x,\lambda y)}}={\frac {M(x,y)}{N(x,y)}}\,.}$
Which to me suggests something about how is we were to rescale each of the axes, the differential equation will remain the same.
Then there is the definition for higher-order equations:
A linear differential equation is called homogeneous if the following condition is satisfied:
If ${\displaystyle \phi (x)}$ is a solution, so is ${\displaystyle c\phi (x)}$, where ${\displaystyle c}$ is an arbitrary (non-zero) constant. Note that in order for this condition to hold, each term in a linear differential equation of the dependent variable y must contain y or any derivative of y. A linear differential equation that fails this condition is called inhomogeneous. A linear differential equation can be represented as a linear operator acting on $y(x)$ where $x$ is usually the independent variable and y is the dependent variable. Therefore, the general form of a linear homogeneous differential equation is
${\displaystyle L(y)=0\,}$
...
It should be noted that the existence of a constant term is a sufficient condition for an equation to be inhomogeneous, as in the above example.
So I get that this definition also has something to do with scaling, although it seems to be only scaling the y.
Is there a connection between these two definitions? Are they referring to the same but just with different order equations?
