Question: Is there a general form for a homogenous PDE of two variables?
In the text I am using, a homogenous PDE:
Has derivatives all of the same order. (1)
Another source which defines a homogenous PDE the same way: https://youtu.be/vxZUqN8SrhE?t=263
However, other texts state that a homogenous DE simply can be written as L(u)=0
In this picture, clearly, not all derivatives are of the same order, which breaks (1) (Taken from this paper)
To add to the confusion, this site claims:
If all the terms of a PDE contains the dependent variable or its partial derivatives then such a PDE is called non-homogeneous partial differential equation
TLDR; What exactly is a homogenous PDE of two variables in its most general form?
In general: A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if $\phi (x)$ is a solution, so is $c\phi (x)$, for any (non-zero) constant $c$. And this is true for homogeneous partial differential equation as well.
And as in your link, it's clearly mentioned : Order of a PDE: The order of the highest derivative term in the equation is called the order of the PDE.
I think the only reason for the confusion is through the video in your question which says the order of every term in the equation should be equal, which is not true in general for the homogeneous partial differential equation.
you can check other examples online also, So far from my understanding I can say that except for that video rest two sources are correct.
Some Trusted Source:
Science direct article, check examples given there
Hindawi Journal
Homogeneous hear equation from wikipedia
notes from ucsb pg-2
Wikiversity
IIT gauhati course notes: