Homogeneous Function

A homogeneous function is one in which all terms have the same degree when the variables are scaled. This homogeneous property makes them useful for solving problems involving physics or engineering concepts. Sometimes, complicated equations can be simplified by applying methods related to homogeneous functions. If f(x, y) is a function and both variables are multiplied by a constant k, then the entire function’s value is multiplied by kn, where n is the degree of the 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. 

To solve a linear homogeneous equation, we use an integrating factor. An integrating factor is a value or expression that simplifies the equation, making it easier to solve. Here are the steps to solve a homogeneous equation: 

Step 1: Identify the standard form of the equation: 
dxdy + P(x)y = 0. 
Step 2: Find the integrating factor (I.F.): 
I.F. = e∫P(x) dx
Step 3: Multiply the entire equation by an integrating factor. This converts the left-hand side into the derivative of (I.F. × y).
Step 4: Integrate both sides with respect to x.
Step 5: Simplify to get the general solution: 
y × e∫P(x) dx = C or y = Ce-∫P(x) dx 

Homogeneous functions are a special type of function where all the terms have the same degree when variables are multiplied by a constant. These functions are very useful in calculus, physics, and economics. Some real-life examples of homogeneous functions are given below: 

• Economics: In economics, homogeneous functions are mainly used in microeconomics and business analysis to model production functions, which show how input factors like labor and capital produce output. 
• Physics: Homogeneous functions help in describing how physical quantities change when the system is scaled. It is used in areas like dynamics, electromagnetism, and thermodynamics. For example, in heat conduction problems, the heat flow equation often uses homogeneous functions to describe temperature distribution across surfaces or materials. 
• Engineering: In civil and mechanical engineering, when analyzing forces and stresses in structures, homogeneous functions are used. Engineers use it in stress-strain analysis; the equations that relate stress and strain in materials under load are often homogeneous, which helps in predicting behavior under different sizes or scaled versions of structures. 
• Computer Graphics: Homogeneous coordinates are used in computer graphics to handle rotations, translations, and scaling of 3D objects. It is used in games, animation, and 3D modeling software.
• Chemistry: Homogeneous functions are used in chemical kinetics to represent rate equations, where the reaction rate depends on concentrations raised to specific powers.