Allied & ICAR Exam Bank • ALLIED_2648 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

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

Question Statement:

For $u(x, y) = x^4 + y^4 - 2 x^2 y^2$, the degree of homogeneity is $n$. The value of $x^2 u_{xx} + 2xy u_{xy} + y^2 u_{yy}$ divided by $u$ is:

Answer Key:

12

Step-by-Step Solution & Explanation

Here $n = 4$. Using the second-order Euler identity: $n(n - 1) = 4 \times 3 = 12$.
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