Question Bank • QB_EM_GATE_007 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

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

Question Statement:

If $u(x, y) = \frac{x^3 + y^3}{\sqrt{x} + \sqrt{y}}$, the value of $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ evaluated at $x = 4, y = 4$ is ________ (round off to 1 decimal place).

Answer Key:

80

Step-by-Step Solution & Explanation

By Euler's Theorem on Homogeneous Functions: Degree of homogeneity $n = 3 - 0.5 = 2.5$. $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n \cdot u(x, y) = 2.5 \times \frac{4^3 + 4^3}{\sqrt{4} + \sqrt{4}} = 2.5 \times \frac{128}{4} = 2.5 \times 32 = 80.0$ Hence, the answer is **80.0**.
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