Allied & ICAR Exam Bank • ALLIED_2645 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

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

Question Statement:

If $u(x, y) = \frac{x^3 + y^3}{x + y}$, then according to Euler's theorem, $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ equals:

Options:

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

Answer Key:

A

Step-by-Step Solution & Explanation

Degree of homogeneity: numerator is degree 3, denominator is degree 1, so $n = 3 - 1 = 2$. By Euler's theorem, $x u_x + y u_y = n u = 2 u$.
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