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GATE 2025 AG – Question 38

Engineering Mathematics · Calculus · 2 marks · Multiple choice

For $u=x^4+y^4+3x^2y^2$,find $\left(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}\right)^{-1}$.

  1. $4u$
  2. 0
  3. $1/(4u)$
  4. $2u$

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Correct answer: (C) $1/(4u)$

Explanation

Every monomial has total degree four. Euler’s theorem for homogeneous functions gives $xu_x+yu_y=4u$. Directly, the terms are $4x^4+6x^2y^2+4y^4+6x^2y^2=4u$. Taking the reciprocal gives **$1/(4u)$,C**,where u is nonzero.

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