# Homogeneous Functions

## Homogeneous

To be **Homogeneous** a function must pass this test:

f(zx, zy) = z^{n} f(x, y)

In other words

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

**and then**can rearrange it to get this: z

^{n}f(x, y)

An example will help:

### Example: x + 3y

**x + 3y**is

**f(x, y)**: f(zx, zy) = z f(x, y)

^{1}f(x, y)

Yes, **x + 3y** is homogeneous!

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

### Example: 4x^{2} + y^{2}

^{2}+ y

^{2}

^{2}+ (zy)

^{2}

^{2}x

^{2}+ z

^{2}y

^{2}

**z**: f(zx, zy) = z

^{2}^{2}(4x

^{2}+ y

^{2})

**4x**is

^{2}+ y^{2}**f(x, y)**: f(zx, zy) = z

^{2}f(x, y)

Yes, **4x ^{2} + y^{2}** is homogeneous.

And its degree is 2.

How about this one:

### Example: x^{3} + y^{2}

^{3}+ y

^{2}

^{3}+ (zy)

^{2}

^{3}x

^{3}+ z

^{2}y

^{2}

**z**: f(zx, zy) = z

^{2}^{2}(zx

^{3}+ y

^{2})

**zx**is NOT

^{3}+ y^{2}**f(x, y)**!

So **x ^{3} + y^{2}** is NOT homogeneous.

And notice that x and y have different powers: x^{3} vs y^{2}. For polynomial functions that is often a good test.

Can it work for functions that are not polynomials? How about this one:

### Example: the function x cos(y/x)

**x cos(y/x)**is

**f(x, y):**f(zx, zy) = z

^{1 }f(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**.

Find out more on Solving Homogeneous Differential Equations.