Allied & ICAR Exam Bank • ALLIED_1020NAT • 2 Marks
Section 1: Engineering Mathematics
Calculus
Subtopic: Partial derivatives
Question Statement:
If $u = \sin^{-1}\left(\frac{x + 2y}{x^2 + y^2}\right)$, then by Euler's Theorem on homogeneous functions, the value of $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ when $u = \pi/6$ is:
Answer Key:
-0.577
Step-by-Step Solution & Explanation
Let $z = \sin(u) = \frac{x+2y}{x^2+y^2}$. Here $z$ is homogeneous of degree $n = 1 - 2 = -1$. By Euler's theorem: $x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = n z = -z$. Since $\frac{\partial z}{\partial x} = \cos(u) \frac{\partial u}{\partial x}$, we get $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = \frac{-z}{\cos(u)} = -\frac{\sin(u)}{\cos(u)} = -\tan(u)$. For $u = \pi/6$: $-\tan(\pi/6) = -\frac{1}{\sqrt{3}} \approx -0.577$.