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GATE 2025 ME – Question 12

Engineering Mathematics · Differential Equations: Heat, wave and Laplace equations · 1 mark · Multiple choice

For the differential equation given below, which one of the following options is correct?

$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \quad\quad 0 \leq x \leq 1, 0 \leq y \leq 1$$

  1. $u = e^{x+y}$ is a solution for all $x$ and $y$
  2. $u = e^x \sin y$ is a solution for all $x$ and $y$
  3. $u = \sin x \sin y$ is a solution for all $x$ and $y$
  4. $u = \cos x \cos y$ is a solution for all $x$ and $y$

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Correct answer: (B) $u = e^x \sin y$ is a solution for all $x$ and $y$

Explanation

For $u = e^x\sin y$ we get $u_{xx} = e^x\sin y$ and $u_{yy} = -e^x\sin y$, which add up to 0. For $e^{x+y}$ the sum is $2e^{x+y}$, and for $\sin x\sin y$ and $\cos x\cos y$ the sum is $-2u$. So only B satisfies Laplace's equation.