Question

Which of the following is not a homogeneous function of \( x \) \text{ and } \( y \)?

• A. \( y^2 - xy \)
• B. \( x - 3y \)
• C. \( \sin^2 \left( \frac{y}{x} \right) + \frac{y}{x} \)
• D. \( \tan x - \sec y \)

Hint:

To determine whether a function is homogeneous, substitute \( x \) and \( y \) with \( tx \) and \( ty \), and check if the function scales by a constant factor \( t^n \).

Answer 1

The Correct Option is D

A function \( f(x, y) \) is homogeneous of degree \( n \) if it satisfies the condition: \[ f(tx, ty) = t^n f(x, y) \] We will now examine each option. Step 1: Test the function \( y^2 - xy \) Substitute \( x = tx \) and \( y = ty \) into the function: \[ f(tx, ty) = (ty)^2 - (tx)(ty) = t^2y^2 - t^2xy = t^2(y^2 - xy) \] Since the function scales by \( t^2 \), it is homogeneous of degree 2. Step 2: Test the function \( x - 3y \) Substitute \( x = tx \) and \( y = ty \): \[ f(tx, ty) = tx - 3(ty) = t(x - 3y) \] Since the function scales by \( t \), it is homogeneous of degree 1. Step 3: Test the function \( \sin^2 \left( \frac{y}{x} \right) + \frac{y}{x} \) Substitute \( x = tx \) and \( y = ty \): \[ f(tx, ty) = \sin^2 \left( \frac{ty}{tx} \right) + \frac{ty}{tx} = \sin^2 \left( \frac{y}{x} \right) + \frac{y}{x} \] Since the function remains unchanged, it is homogeneous of degree 1. Step 4: Test the function \( \tan x - \sec y \) Substitute \( x = tx \) and \( y = ty \): \[ f(tx, ty) = \tan(tx) - \sec(ty) \] This function does not scale by any power of \( t \). Therefore, it is not homogeneous. Thus, the function \( \tan x - \sec y \) is not homogeneous, and the correct answer is (D).