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For the differential equation given below, which one of the following options is correct?
$$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 \quad 0 \leq x \leq 1,0 \leq y \leq 1 $$

  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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