Homogeneous Function — Definition, Formula & Examples

A homogeneous function is one where multiplying every input by a constant

t

is equivalent to multiplying the entire output by

t^n

, where

n

is called the degree of homogeneity.

A function $f(x_1, x_2, \ldots, x_k)$ is homogeneous of degree $n$ if for all $t > 0$ , $f(tx_1, tx_2, \ldots, tx_k) = t^n f(x_1, x_2, \ldots, x_k)$ . The exponent $n$ can be any real number, including zero or negative values.

Key Formula

f(tx_1, tx_2, \ldots, tx_k) = t^n \, f(x_1, x_2, \ldots, x_k)

Where:

$f$ = The function being tested for homogeneity
$t$ = A positive scaling factor applied to all inputs
$n$ = The degree of homogeneity
$x_i$ = The input variables of the function

How It Works

To check whether a function is homogeneous, replace every variable

x_i

with

tx_i

and simplify. If you can factor out

t^n

and recover the original function, it is homogeneous of degree

n

. This property is essential for solving first-order homogeneous ODEs, where you substitute

y = vx

to reduce the equation to a separable form. Euler's theorem for homogeneous functions also connects the function to its partial derivatives, which appears frequently in thermodynamics and economics.

Worked Example

Problem: Determine whether f(x, y) = x³ + 2x²y − y³ is homogeneous, and if so, find its degree.

Replace each variable: Substitute

tx

for

x

and

ty

for

y

f(tx, ty) = (tx)^3 + 2(tx)^2(ty) - (ty)^3

Expand and factor: Expand each term and factor out the common power of

t

= t^3 x^3 + 2t^3 x^2 y - t^3 y^3 = t^3(x^3 + 2x^2 y - y^3)

Identify the degree: The result equals

t^3 f(x,y)

, so the function is homogeneous of degree 3.

f(tx, ty) = t^3 \, f(x, y)

Answer: The function is homogeneous of degree 3.

Why It Matters

Recognizing a homogeneous function lets you solve certain first-order ODEs by reducing them to separable equations via the substitution

y = vx

. Euler's homogeneous function theorem (

x_1 f_{x_1} + \cdots + x_k f_{x_k} = nf

) is used in thermodynamics to derive properties of extensive variables and in economics to analyze constant-returns-to-scale production functions.

Common Mistakes

Mistake: Confusing a homogeneous function with a homogeneous ODE (one where the right-hand side is zero).

Correction: These are different uses of the word 'homogeneous.' A homogeneous function satisfies the scaling property

f(t\mathbf{x}) = t^n f(\mathbf{x})

. A homogeneous linear ODE simply means there is no forcing term. Always check context.