Homogeneous Functions

To be Homogeneous a function must pass this test:

f(zx,zy) = zⁿf(x,y)

In other words

Homogeneous is when we can take a function:f(x,y)

multiply each variable by z:f(zx,zy)

and then can rearrange it to get this:zⁿf(x,y)

An example will help:

Example: x + 3y

Start with:f(x,y) = x + 3y

Multiply each variable by z:f(zx,zy) = zx + 3zy

Let’s rearrange it by factoring out z:f(zx,zy) = z(x + 3y)

And x + 3y is f(x,y):f(zx,zy) = zf(x,y)

Which is what we wanted, with n=1:f(zx,zy) = z¹f(x,y)

Yes it is homogeneous!

The value of n is called the degree. So in that example the degree is 1.

Example: 4x² + y²

Start with:f(x,y) = 4x² + y²

Multiply each variable by z:f(zx,zy) = 4(zx)² + (zy)²

Which is:f(zx,zy) = 4z²x² + z²y²

Factoring out z²:f(zx,zy) = z²(4x² + y²)

And 4x² + y² is f(x,y):f(zx,zy) = z²f(x,y)

Yes 4x² + y² is homogeneous.

And its degree is 2.

How about this one:

Example: x³ + y²

Start with:f(x,y) = x³ + y²

Multiply each variable by z:f(zx,zy) = (zx)³ + (zy)²

Which is:f(zx,zy) = z³x³ + z²y²

Factoring out z²:f(zx,zy) = z²(zx³ + y²)

But zx³ + y² is NOT f(x,y)!

So x³ + y² is NOT homogeneous.

And notice that x and y have different powers: x³ but y² which, for polynomial functions, is often a good test.

But not all functions are polynomials. How about this one:

Example: the function x cos(y/x)

Start with:f(x,y) = x cos(y/x)

Multiply each variable by z:f(zx,zy) = zx cos(zy/zx)

Which is:f(zx,zy) = zx cos(y/x)

Factoring out z:f(zx,zy) = z(x cos(y/x))

And x cos(y/x) is f(x,y):f(zx,zy) = z¹f(x,y)

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.