Homogeneous Function
2026-02-28 09:56 Diff
https://brightchamps.com/en-us/math/algebra/homogeneous-function

Euler’s theorem states that, if f(x, y, z) is a homogeneous function of degree n, then:
x∂f∂x + y∂f∂y + z∂f∂z = n f(x, y, z)
This means that if we multiply each variable by its partial derivative and then add them, we will get the degree n times the original function. 

When a function is homogeneous, scaling its variable by a number t changes the whole function by tn. Euler’s theorem is a rule that comes from this scaling behavior. The steps given below explain Euler’s theorem:

Step 1: Assume f(x, y, z) is homogeneous of degree n. 
f(tx, ty, tz) = tn f(x, y, z)

Step 2: Differentiate both sides of the equation with respect to t, applying the derivative rule to each term.

Step 3: Apply the chain rule to the left-hand side: 
x∂f∂x + y∂f∂y + z∂f∂z

Step 4: Differentiate the right-hand side,
ntn - 1f(x, y, z)

Step 5: Set t = 1, and you get,
x∂f∂x + y∂f∂y + z∂f∂z = n f(x, y, z)

Hence, Euler's theorem is proved.

Homogeneous Differential Equation from Homogeneous Function

A homogeneous differential equation whose right-hand side is made from a homogeneous function of x and y. It can be written as:


dydx = f(x, y)
If f(x, y) is a homogeneous function, then the equation is referred to as a homogeneous differential equation.


A function f(x, y) is called homogeneous of degree n if we can write it as:


f(x, y) = xn × gyx or f(x, y) = yn × gxy

The homogeneous function depends on the ratio of y to x or x to y.

A differential equation is called homogeneous when it can be written in the form of,
dydx = g(x, y)
This means the right side of the equation depends only on the ratio yx. The steps to solve it are provided below.

Step 1: We use a new variable: v = yx or y = v . x

Step 2: If y = v . x, we use the product rule to find dydx

dydx = v + xdvdx

Step 3: Substitute the value dydx into the given equation:

v + xdvdx = g(v)

Step 4: Rearrange to isolate dvdx

xdvdx = g(v) - v

Step 5: Separate the variables to rewrite the equation to separate v and x:

dvg(v) - v = dxx

Step 6: Integrate both sides,

1g(v) - vdv =1xdx + C

  Here, C is the constant.

Step 7: Since v = yx, we substitute it back into the solution to get the answer in terms of x and y.
If the equation is dxdy = f(x, y) is homogeneous, we use x = vy and solve it the same way.