Question Bank • QB_EM_141 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Homogeneous function – Euler's theorem on homogeneous functions

Question Statement:

If $u(x, y) = \frac{x^4 + y^4}{x + y}$, calculate the value of $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ when $u = 8$.

Answer Key:

24

Step-by-Step Solution & Explanation

The degree of homogeneity $n$ is: $u(tx, ty) = \frac{t^4(x^4 + y^4)}{t(x + y)} = t^3 u(x, y) \implies n = 3$ By Euler's theorem on homogeneous functions: $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n u = 3 \times 8 = 24$
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