I'm studying ODE.
Over my math education career, I've occasionally encountered reference to homogeneous equations. Now, in ODE, I'm learning how to solve differential equations that are homogeneous. I've learned how to do this technique and how to recognize a homogeneous equation (and its order).
What I'm not so sure about is the significance of a homogeneous equation. Is it just a type of equation that lends itself to particular solutions? Does the fact that it represents an equation dependent on some ratio between x and y matter? Is there some physical significance to homogeneous equations?
Definition of Homogeneous ODE,
copy from https://mathworld.wolfram.com/HomogeneousOrdinaryDifferentialEquation.html :
A linear ordinary differential equation of order $n$ is said to be homogeneous if it is of the form $$a_n(x)y^ {(n)}+a_{n-1}(x)y^{(n-1)}+...+a_1(x)y'+a_0(x)y=0$$
where $y'=\frac{dy}{dx}$, i.e., if all the terms are proportional to a derivative of $y$ (or $y$ itself) and there is no term that contains a function of $x$ alone.
However, there is also another entirely different meaning for a first-order ordinary differential equation. Such an equation is said to be homogeneous if it can be written in the form
$$\frac{dy}{dx}=F\left(\frac{y}{x} \right).$$
Such equations can be solved in closed form by the change of variables $u=\frac{y}{x}$ which transforms the equation into the separable equation $$\frac{dx}{x}=\frac{du}{F(u)-u}.$$
NOTE : The bounty is not for me. It should be given to the author of the above citation.