Subtopic: Homogeneous function – Euler's theorem on homogeneous functions
Question Statement:
If $u(x, y) = \frac{x^3 + y^3}{x + y}$, the value of $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ evaluated at $(x=2, y=3)$ is ________ (answer in integer).
Answer Key:
14
Step-by-Step Solution & Explanation
Determine the degree of homogeneity of $u(x, y)$:
$u(tx, ty) = \frac{t^3 x^3 + t^3 y^3}{tx + ty} = t^2 \frac{x^3 + y^3}{x + y} = t^2 u(x, y)$
So $u$ is a homogeneous function of degree $n = 2$.
By Euler's theorem for homogeneous functions:
$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n \cdot u(x, y) = 2 u(x, y)$
At $(x=2, y=3)$:
$u(2, 3) = \frac{2^3 + 3^3}{2 + 3} = \frac{8 + 27}{5} = \frac{35}{5} = 7$
Therefore:
$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 2 \times 7 = 14$