Question

Determining if a Function Is Homogeneous In Exercises \(67-74,\) determine whether the function is homogeneous, and if it is, determine its degree. A function \(f(x, y)\) is homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y)\) $$ f(x, y)=\frac{x^{2} y^{2}}{\sqrt{x^{2}+y^{2}}} $$

Short answer

The function \(f(x, y)=\frac{x^{2} y^{2}}{\sqrt{x^{2}+y^{2}}\) is homogeneous of degree 3.

Step-by-step solution

Homogeneity Check

Substitute \(x = t*x\) and \(y = t*y\) into the function \(f(x, y)=\frac{x^{2} y^{2}}{\sqrt{x^{2}+y^{2}}\) to check if it obeys the homogeneity condition. This gives us: \(f(t x, t y)=\frac{(t*x)^{2} (t*y)^{2}}{\sqrt{(t*x)^{2}+(t*y)^{2}}}\)

Simplify Equation

Simplify the equation obtained in step 1 to observe if it follows the homogeneity pattern. The resulting equation is \(f(t x, t y)=\frac{t^{4} x^{2} y^{2}}{t \sqrt{x^{2}+y^{2}}} = t^{3} f(x,y)\)

Conclude

Observe that \(t^{3} f(x,y)\) fits the definition of a homogeneous function given as \(f(t x, t y)=t^{n} f(x, y)\). So, the function is homogeneous.

Key concepts

Degree of Homogeneity

The degree of homogeneity is a key characteristic of a homogeneous function. A multivariable function \( f(x, y) \) is homogeneous of degree \( n \) if, when all input variables are multiplied by a scalar \( t \), the function itself is multiplied by \( t^n \). For example, if \( f(tx, ty) = t^n f(x, y) \), then the function is homogeneous of degree \( n \).

• In the context of our example, the function \( f(x, y) = \frac{x^2 y^2}{\sqrt{x^2 + y^2}} \) is homogeneous.
• By substituting \( tx \) and \( ty \) into the function, we derive \( f(tx, ty) = \frac{(t^2 x^2)(t^2 y^2)}{\sqrt{t^2 x^2 + t^2 y^2}} = \frac{t^4 x^2 y^2}{t \sqrt{x^2 + y^2}} = t^3 f(x, y) \).
• Thus, the function is homogeneous of degree \( 3 \) as shown by the symmetry and scale behavior observed in the final expression \( t^3 f(x, y) \).

Determining the degree of homogeneity is crucial in contexts such as economics, where functions often represent production outputs and understanding their scale dependencies lets economists predict outcomes under varying input conditions.

Multivariable Functions

Multivariable functions involve more than one variable. They are essential in fields like physics, engineering, and economics, where systems depend on multiple factors. Consider \( f(x, y) \) as a simple example, where the output of the function varies with both \( x \) and \( y \).

• These functions can often be represented as surfaces in three-dimensional space, with \( f(x, y) \) giving the 'height' above the \( (x, y) \) plane.
• In the case of a homogeneous multivariable function, these surfaces have the property that rescaling the input variables by a common factor results in rescaling the function output by a predictable factor.
• This property makes such functions easier to analyze, as they show consistent behavior under scaling, simplifying both theoretical and practical applications.

Functions like \( f(x, y) = \frac{x^2 y^2}{\sqrt{x^2 + y^2}} \), where both variables influence the outcome, are prevalent in modeling real-world phenomena.

Differential Equations

Differential equations describe the relationship between functions and their derivatives. They are fundamental in describing natural phenomena, from the orbits of planets to population dynamics. Homogeneous functions often appear in differential equations, where their scaling properties simplify the process of finding solutions.

• In the case of a homogeneous differential equation, every term is of the same degree of homogeneity. This uniformity leads to a clear strategy for separation of variables and finding particular solutions.
• The degree of a function in a differential equation can help analysts predict how a system responds to scaling and how complex dynamics can be decomposed into simpler, solvable parts.
• For example, if a system follows a homogeneous differential equation, knowing the degree of homogeneity can directly indicate the form and behavior of solutions.

Understanding how homogeneous functions interact with differential equations unleashes powerful analytical techniques for solving complex problems across scientific disciplines.