Question Bank • QB_SUB_EM_0042 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

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

Question Statement:

If $u(x, y) = x^3 + y^3 + x y^2$ is a homogeneous function of degree $n = 3$, the value of $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ when $u = 5.0$ is:

Answer Key:

15.0

Step-by-Step Solution & Explanation

By Euler's theorem on homogeneous functions of degree $n$: $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n u$ Here $n = 3$ and $u = 5.0$: $x u_x + y u_y = 3 \times 5.0 = 15.00$
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