Subtopic: Homogeneous function – Euler's theorem on homogeneous functions
Question Statement:
If $u = \sin^{-1}\left( \frac{x^3 + y^3}{x + y} \right)$, calculate the value of $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ when $u = \frac{\pi}{6}$.
Answer Key:
1.155
Step-by-Step Solution & Explanation
Let $z = \sin(u) = \frac{x^3 + y^3}{x + y}$.
$z$ is a homogeneous function of degree $n = 3 - 1 = 2$.
By Euler's theorem for homogeneous functions:
$x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = n z = 2 z$
Since $z = \sin(u)$:
$\frac{\partial z}{\partial x} = \cos(u) \frac{\partial u}{\partial x}, \quad \frac{\partial z}{\partial y} = \cos(u) \frac{\partial u}{\partial y}$
$\cos(u) \left( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} \right) = 2 \sin(u)$
$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 2 \tan(u)$
For $u = \frac{\pi}{6}$ ($30^\circ$):
$2 \tan(30^\circ) = 2 \times \frac{1}{\sqrt{3}} = \frac{2}{1.732} \approx 1.1547$