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:

homogeneous equation

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 onSolving Homogeneous Differential Equations.