Question Bank • QB_EM_ADV_032 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Homogeneous function – Euler's theorem on homogeneous functions

Question Statement:

The degree of homogeneity $n$ of the rational function $u(x, y) = \frac{x^4 + y^4}{x^2 y + x y^2}$ is:

Options:

  • (A) $1$
  • (B) $2$
  • (C) $3$
  • (D) $4$

Answer Key:

A

Step-by-Step Solution & Explanation

Substitute $x \to tx$ and $y \to ty$: $u(tx, ty) = \frac{(tx)^4 + (ty)^4}{(tx)^2(ty) + (tx)(ty)^2} = \frac{t^4(x^4 + y^4)}{t^3(x^2 y + x y^2)} = t^{4-3} u(x, y) = t^1 u(x, y)$ Hence the degree of homogeneity is $n = 1$.
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