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 defined as homogeneous of degree \( n \) if \( f(tx, ty) = t^n f(x, y) \). We will now evaluate each option. Step 1: Consider the function \( y^2 - xy \). Upon substituting \( x = tx \) and \( y = ty \), we get \( f(tx, ty) = (ty)^2 - (tx)(ty) = t^2y^2 - t^2xy = t^2(y^2 - xy) \). This indicates homogeneity of degree 2. Step 2: Consider the function \( x - 3y \). Substituting \( x = tx \) and \( y = ty \) yields \( f(tx, ty) = tx - 3(ty) = t(x - 3y) \). This indicates homogeneity of degree 1. Step 3: Consider the function \( \sin^2 \left( \frac{y}{x} \right) + \frac{y}{x} \). Substituting \( x = tx \) and \( y = ty \) results in \( f(tx, ty) = \sin^2 \left( \frac{ty}{tx} \right) + \frac{ty}{tx} = \sin^2 \left( \frac{y}{x} \right) + \frac{y}{x} \). Since the function remains unchanged, it is homogeneous of degree 1. Step 4: Consider the function \( \tan x - \sec y \). Substituting \( x = tx \) and \( y = ty \) gives \( f(tx, ty) = \tan(tx) - \sec(ty) \). This function does not exhibit scaling by any power of \( t \), and thus it is not homogeneous. Therefore, \( \tan x - \sec y \) is not homogeneous, leading to option (D) as the correct choice.