Subtopic: Homogeneous function – Euler's theorem on homogeneous functions
Question Statement:
If $u = \ln\left(\frac{x^4 + y^4}{x + y}\right)$, the value of the expression $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ is ________ (in integer).
Answer Key:
3
Step-by-Step Solution & Explanation
1. Let $z = e^u = \frac{x^4 + y^4}{x + y}$.
2. The degree of homogeneity of $z$ is $n = 4 - 1 = 3$.
3. By Euler's theorem on homogeneous functions:
$x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 3 z$
4. Since $z = e^u$, $\frac{\partial z}{\partial x} = e^u \frac{\partial u}{\partial x} = z \frac{\partial u}{\partial x}$ and $\frac{\partial z}{\partial y} = z \frac{\partial u}{\partial y}$.
5. Dividing both sides by $z$:
$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 3$