homogeneous: what does it mean?

• Google "homogeneous function" and you get this page:
  http://mathworld.wolfram.com/HomogeneousFunction.html
  but it only considers the case of 2 independent variables
  but Wikipedia gives more examples on its page:
  http://en.wikipedia.org/wiki/Homogeneous_function
   
• So go to an on-line dictionary like Merriam-Webster and find:
  http://www.merriam-webster.com/dictionary/homogeneous?show=0 [among top 1% lookups in 2013!]
  namely:
  Main Entry: ho·mo·ge·neous
  Pronunciation: -'jE-nE-&s, -ny&s [hear it by clicking on audio icon!]
  Function: adjective
  Etymology: Medieval Latin homogeneus, homogenus, from Greek homogenEs, from hom- + genos kind --
  1 : of the same or a similar kind or nature
  2 : of uniform structure or composition throughout <a culturally homogeneous neighborhood>
  3 : having the property that if each variable is replaced by a constant times that variable the constant can be factored out : having each term of the same degree if all variables are considered <a homogeneous equation>
  First Known Use: 1641

Bingo: the last definition is exactly what we want. If we write that definition in math language for a function of one independent variable we get:
f(t x) = tⁿ f(x)
or for two independent variables:
f(t x, t y) = tⁿ f(x, y) .
The exponent n is called the degree of the homogeneous function.

Obvious examples of this would be power functions in the first case of a single variable:
f(x) =  a xⁿ since by rules of exponents: f(t x) =  a (t x)ⁿ =  a t ⁿ xⁿ = t ⁿ f(x)

In the second case of two independent variables, an example of a homogeneous function of degree two would be a quadratic function with no linear or constant terms: f(x ,y) = a x² + b y² + c x y , while a linear homogeneous function (the degree has to be 1) would be a linear function with no constant term:  f(x ,y) = a x + b y, as opposed to a linear function which allows a constant term:   f(x ,y) = a x + b y +c. The constant term breaks the homogeneity rule, so for linear functions,  it is important to distinguish the homogeneous and nonhomogenous cases. In general any linear combination of products of powers of the variables such that the sum of the exponents in each term is fixed will define a homogenous function, or even quotients or powers of such functions, like the function f(x ,y) = xy²/(x² + y²) (this example has degree 1, as the quotient of homogeneous functions of degree 3 and 2 respectively), while f(x ,y) = xy²/(x² + y⁴) is not homogeneous and for exactly this reason is a great example of why limits are conceptually more complicated for multivariable functions than those of a single variable.

For functions of a single variable:
linear homogeneous: f(x) =  a x ,
linear nonhomogeneous:  f(x) =  a x + b .

For functions of a two variables:
linear homogeneous: f(x,y) =  a x + b y ,
linear nonhomogeneous:  f(x,y) =  a x + b y + c.

For an unknown y and its derivative y ' as the two variables in our definition of a homogeneous function of two variables:
a y ' + b y + c = 0 is a linear nonhomogeneous function of  the two variables set equal to zero, while
a y ' + b y = 0 is a linear homogeneous function set equal to zero, respectively called a linear (nonhomogeneous) differential equation and a linear homogeneous differential equation.

Linear Superposition
One can always add any solution of the linear homogeneous equation to solutions of the corresponding linear nonhomogeneous equation and still get solutions of the latter equation, so one has a restricted principle of superposition applying to solutions of the pair of equations together, namely:
nonhomogeneous linear DE:  a y_p ' + b y_p + c = 0  (p for particular solution)
      homogeneous linear DE:  a y_h ' + b y_h = 0         (h for homogeneous solution)
Adding these together and using the sum rule for differentiation to collect together the two derivative terms:
a (y_p + y_h )' + b(y_p + y_h ) + c = 0
so any solution of the homogeneous equation can be added to a particular solution of the nonhomogeneous equation to get another solution of the latter equation.

In particular if we find the general solution of a homogeneous linear differential equation and add to it any particular solution of a corresponding nonhomogeneous linear differential equation, we get the general solution of the latter differential equation. This gives us a divide and conquer strategy for finding that general solution in two steps.