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GATE 2025 CE (CE1) – Question 28

Engineering Mathematics · PDE: Fourier series, separation of variables, diffusion, wave and Laplace equations · 1 mark · Multiple select

Which of the following equations belong/belongs to the class of second-order, linear, homogeneous partial differential equations:

  1. $\frac{\partial^2 u}{\partial t^2} = c^2\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) + xy$
  2. $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0$
  3. $\frac{\partial u}{\partial t} = c\frac{\partial u}{\partial x}$
  4. $\left(\frac{\partial^2 u}{\partial t^2}\right)^2 = c^2\frac{\partial^2 u}{\partial x^2}$

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Correct answer: (B) $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0$

Explanation

Option A has a term $xy$ that does not contain $u$, so it is not homogeneous. Option C is first order. Option D has the second derivative squared, so it is not linear. Option B, Laplace's equation in three dimensions, is second order, linear and homogeneous.