Why are homogeneous and non-homogeneous first order differential equations called homogeneous an vice versa?

Question

So I have recently been studying differential equations and I am extremely confused as to why the properties of homogeneous and non-homogeneous equations were given those names. It seems to have very little to do with their properties are. Is there some reason for their naming scheme?

Answer 1

A function is homogeneous of degree $n$ if

$$f(kx,ky) = k^nf(x,y).$$

This is a two variable example, but you could have more.

A homogeneous equation is one that might look like

$$f(x,y,z) =0$$

where $f$ is a homogeneous function. For instance

$$x^n+y^n-z^n =0$$.

What's special about a homogeneous equation is that you can multiply all the variables by a constant and it doesn't really change the equation. If you multiply $x$, $y$ and $z$ by a constant $k$ in the last equation, you can factor out a $k^n$ and divide it out. In a homogeneous equation, if $(x,y,z)$ is a solution, then so is $(kx,ky,kz).$

So now, a homogeneous differential equation is one where you can multiply a solution by a constant and it's still a a solution. When you solve a homogeneous linear equation and find a solution like $y=e^{2t}$, then you know that $y=ke^{2t}$ is also a solution. You get infinite solutions for the price of one.

Non-homogeneous equations don't have that nice property.