Question

Which of the following functions is/are homogeneous?

• A. \(xcot^{-1}(\frac{y}{x})\)
• B. \(\sqrt{\frac{x}{y} }+ \frac{3x}{y} +7\)
• C. \(\frac{x^3+y^3}{ 3x+4y}\)
• D. \(3x^5y+2x^2y^4-3x^3y^4\)

Answer 1

The Correct Option is A, B, C

To determine which functions are homogeneous, we need to recall that a function \(f(x, y)\) is homogeneous of degree \(n\) if, for every scalar \(t\), the relation \( f(tx, ty) = t^n f(x, y) \) holds.

Consider the function \( f(x, y) = x \cot^{-1} \left(\frac{y}{x}\right) \).

Substitute \(tx\) and \(ty\) for \(x\) and \(y\):

\(f(tx, ty) = tx \cot^{-1} \left(\frac{ty}{tx}\right) = tx \cot^{-1} \left(\frac{y}{x}\right)\)

We find that:

\(f(tx, ty) = t \cdot x \cot^{-1} \left(\frac{y}{x}\right) = t \cdot f(x, y)\)

This function is homogeneous of degree 1.

Consider the function \( f(x, y) = \sqrt{\frac{x}{y} } + \frac{3x}{y} + 7 \).

Substitute \(tx\) and \(ty\):

\(f(tx, ty) = \sqrt{\frac{tx}{ty}} + \frac{3(tx)}{ty} + 7 = \sqrt{\frac{x}{y}} + \frac{3x}{y} + 7\)

This shows that:

\(f(tx, ty) = f(x, y)\)

This function is homogeneous of degree 0.

Consider the function \( f(x, y) = \frac{x^3 + y^3}{3x + 4y} \).

Substitute \(tx\) and \(ty\):

\(f(tx, ty) = \frac{(tx)^3 + (ty)^3}{3(tx) + 4(ty)} = \frac{t^3(x^3 + y^3)}{t(3x + 4y)} = t^2 \cdot \frac{x^3 + y^3}{3x + 4y} = t^2 f(x, y)\)

This function is homogeneous of degree 2.

Consider the function \( f(x, y) = 3x^5y + 2x^2y^4 - 3x^3y^4 \).

Substitute \(tx\) and \(ty\):

\(f(tx, ty) = 3(tx)^5(ty) + 2(tx)^2(ty)^4 - 3(tx)^3(ty)^4\)

\(= 3t^6x^5y + 2t^6x^2y^4 - 3t^7x^3y^4\)

This function does not simplify to \(t^n f(x, y)\) for a single degree \(n\), so it is not homogeneous.

Therefore, the functions which are homogeneous are:

• \( x \cot^{-1} (\frac{y}{x}) \) (Degree 1)
• \( \sqrt{\frac{x}{y} }+ \frac{3x}{y} +7 \) (Degree 0)
• \( \frac{x^3+y^3}{ 3x+4y} \) (Degree 2)