To be Homogeneous a function must pass this test:
f(zx, zy) = zn f(x, y)
In other words
An example will help:
Example: x + 3y
Yes, x + 3y is homogeneous!
The value of n is called the degree. So in that example the degree is 1.
Example: 4x2 + y2
Yes, 4x2 + y2 is homogeneous.
And its degree is 2.
How about this one:
Example: x3 + y2
So x3 + y2 is NOT homogeneous.
And notice that x and y have different powers: x3 vs y2. For polynomial functions that is often a good test.
But not all functions are polynomials. How about this one:
Example: the function x cos(y/x)
So x cos(y/x) is homogeneous, with degree of 1.
Notice that (y/x) is "safe" because (zy/zx) cancels back to (y/x)
Homogeneous, in English, means "of the same kind"
For example "Homogenized Milk" has the fatty parts spread evenly through the milk (rather than having milk with a fatty layer on top.)
Homogeneous applies to functions like f(x), f(x, y, z) etc. It is a general idea.
Homogeneous Differential Equations
A first order Differential Equation is homogeneous when it can be in this form:
In other words, when it can be like this:
M(x, y) dx + N(x, y) dy = 0
And both M(x, y) and N(x, y) are homogeneous functions of the same degree.
Find out more on Solving Homogeneous Differential Equations.