What is a homogeneous equation function

Definition: homogeneous functions
A function f (x, y) is called a homogeneous function of degree n,
if you can write it in the following form:

annotation: The Degree of homogeneity of a function
has nothing to do with that Degree of a differential equation to do.

Explanation and examples
A function f (x, y) is homogeneous if it can be written as a product,
where one factor is a power of x and the other factor is a function
of y / x (i.e. a function whose argument is y / x). In the example y / x
the argument of the sine function:


In this example, y / x is the argument of the power function:

The function of y / x can consist of several sub-functions,
in the example from the sine function and an exponential function:

The power of x can also be x0= 1, i.e. omitted.
The degree of homogeneity is then zero:

The second factor can also be a function of x / y instead of y / x,
because you can always write x / y as a double fraction:

Recognizing the homogeneity by factoring out
Particularly in the case of completely rational functions, the homogeneity can only be recognized when
if one excludes a power of x. example

On the next page we will therefore specify a rule that allows you to do without
previous bracketing can very easily check whether a completely rational function
is homogeneous and what degree of homogeneity has, i.e. what power xn man
can exclude.