Problem 250 - Calculus 3

Question

For the following exercises, use this information: A function \(f(x, y)\) is said to be homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) For all homogeneous functions of degree \(n, \quad\) the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\) . Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\) $$ f(x, y)=x^{2} y-2 y^{3} $$

Step-by-Step Solution

Answer

The function is homogeneous of degree 3, and the equation is verified for this degree.

Step 1: Verify Homogeneity of the Function

To assess whether the function \( f(x, y) = x^2y - 2y^3 \) is homogeneous, we apply the definition: a function is homogeneous of degree \( n \) if \( f(tx, ty) = t^n f(x, y) \). We substitute \( tx \) and \( ty \) in place of \( x \) and \( y \) in the function:\[ f(tx, ty) = (tx)^2(ty) - 2(ty)^3 = t^2 \cdot x^2 \cdot t \cdot y - 2t^3y^3 = t^3x^2y - 2t^3y^3 \]Simplifying this gives:\[ t^3(x^2y - 2y^3) = t^3f(x, y) \]Therefore, \( n = 3 \). The function is homogeneous of degree 3.

Step 2: Calculate Partial Derivatives

To verify the given equation, find the partial derivatives of \( f(x, y) = x^2y - 2y^3 \). First, calculate \( \frac{\partial f}{\partial x} \):\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2y - 2y^3) = 2xy \]Next, calculate \( \frac{\partial f}{\partial y} \):\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^2y - 2y^3) = x^2 - 6y^2 \]

Step 3: Verify Given Equation

The given equation to verify is:\[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = nf(x, y) \]Substitute the partial derivatives:\[ x(2xy) + y(x^2 - 6y^2) = 2x^2y + x^2y - 6y^3 = 3x^2y - 6y^3 \]Since \( n = 3 \), the right side is:\[ 3(x^2y - 2y^3) = 3x^2y - 6y^3 \]Both sides are equal: \( 3x^2y - 6y^3 = 3x^2y - 6y^3 \).Therefore, the equation is satisfied.

Key Concepts

Partial Derivatives

Partial derivatives are essential when dealing with functions that have multiple variables. Consider a function of two variables, like in our exercise, where we have \( f(x, y) = x^2y - 2y^3 \). Here, the function depends on both \( x \) and \( y \), and a partial derivative provides the rate at which the function changes as one variable changes while keeping the other constant.
To compute the partial derivative of \( f \) with respect to \( x \), denoted as \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant. Therefore:\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2y - 2y^3) = 2xy \] For the partial derivative with respect to \( y \), represented as \( \frac{\partial f}{\partial y} \), treat \( x \) as a constant:\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^2y - 2y^3) = x^2 - 6y^2 \]

• Use \( \frac{\partial f}{\partial x} \) when assessing changes due to \( x \) variation.
• Use \( \frac{\partial f}{\partial y} \) when assessing changes due to \( y \) variation.

Understanding partial derivatives is vital for analyzing how functions behave locally based on each independent variable. In problems involving homogeneity, partial derivatives play a crucial role in verifying certain mathematical properties.

Degree of Homogeneity

The degree of homogeneity is a measure that indicates how a function scales when its variables are multiplied by a constant factor. A function \( f(x, y) \) is said to be homogeneous of degree \( n \) if when each variable is replaced by its corresponding scaled version, it satisfies:\[ f(tx, ty) = t^n f(x, y) \]This tells us that scaling the input variables by \( t \) results in the output being scaled by \( t^n \).
Let's verify this for the function given in our exercise. If we substitute \( tx \) and \( ty \) for \( x \) and \( y \), respectively, we get:\[ f(tx, ty) = (tx)^2 (ty) - 2 (ty)^3 = t^3 (x^2y - 2y^3) = t^3 f(x, y) \]From this calculation, we can see that the function is homogeneous of degree \( 3 \). This property is particularly useful in simplifying complex equations and analyzing systems where proportional scaling occurs, like in economics or physics.

Verification of Homogeneous Property

Verification involves ensuring that a homogeneous function respects the well-known identity that connects its degree of homogeneity with partial derivatives. For a homogeneous function of degree \( n \), the expression\[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y) \]should hold true. Using this identity helps confirm the consistency of the function's defined scaling behavior.
To verify, substitute the partial derivatives from our function: \( \frac{\partial f}{\partial x} = 2xy \) and \( \frac{\partial f}{\partial y} = x^2 - 6y^2 \).Replacing in the identity, we have:\[ x(2xy) + y(x^2 - 6y^2) = 2x^2y + x^2y - 6y^3 = 3x^2y - 6y^3 \]Verify this matches \( n f(x, y) \) where \( n = 3 \):\[ 3(x^2y - 2y^3) = 3x^2y - 6y^3 \]Both sides match, confirming the function satisfies the homogeneous property. Such verification is crucial in mathematical modeling to ensure equations adhere to theoretical constructs.