# 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)**

*multiply each variable by z:*

**f(zx,zy)**

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

^{n}f(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) = z**

^{1}f(x,y)Yes it is homogeneous!

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

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

*Start with:*

**f(x,y) = 4x**

^{2}+ y^{2}*Multiply each variable by z:*

**f(zx,zy) = 4(zx)**

^{2}+ (zy)^{2}*Which is:*

**f(zx,zy) = 4z**

^{2}x^{2}+ z^{2}y^{2}*Factoring out*

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

^{2}(4x^{2}+ y^{2})*And*

**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}

*Start with:*

**f(x,y) = x**

^{3}+ y^{2}*Multiply each variable by z:*

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

^{3}+ (zy)^{2}*Which is:*

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

^{3}x^{3}+ z^{2}y^{2}*Factoring out*

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

^{2}(zx^{3}+ y^{2})*But*

**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}**but

**y**which, for polynomial functions, is often a good test.

^{2}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) = 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 onSolving Homogeneous Differential Equations.