Allied & ICAR Exam Bank • ALLIED_2646NAT • 2 Marks
Section 1: Engineering Mathematics
Calculus
Subtopic: Homogeneous function – Euler's theorem on homogeneous functions
Question Statement:
Let $u = \sin^{-1}\left(\frac{x + 2y}{\sqrt{x} + \sqrt{y}}\right)$. The value of $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ at any point is $k \tan u$. The value of $k$ is:
Answer Key:
0.5
Step-by-Step Solution & Explanation
Let $v = \sin u = \frac{x + 2y}{\sqrt{x} + \sqrt{y}}$. The degree of $v$ is $1 - 0.5 = 0.5$. By Euler's theorem for $v$: $x v_x + y v_y = 0.5 v$. Since $v_x = \cos u \cdot u_x$, this yields $\cos u (x u_x + y u_y) = 0.5 \sin u \implies x u_x + y u_y = 0.5 \tan u$. Thus $k = 0.5$.