Allied & ICAR Exam Bank • ALLIED_2650 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

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

Question Statement:

If $u = \ln\left(\frac{x^4 + y^4}{x + y}\right)$, then $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ is a constant. That constant value is:

Answer Key:

3

Step-by-Step Solution & Explanation

Let $v = e^u = \frac{x^4 + y^4}{x + y}$. The degree of $v$ is $4 - 1 = 3$. Then $x v_x + y v_y = 3 v$. Since $v_x = e^u u_x = v u_x$, we get $v(x u_x + y u_y) = 3v \implies x u_x + y u_y = 3$.
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