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Consider the following partial differential equation $\text{(PDE)}$

$$a \frac{ \partial^{2} f(x ,y)}{\partial x^{2}} + b \frac{ \partial^{2} f(x ,y)}{\partial y^{2}} = f(x, y),$$

where $a$ and $b$ are distinct positive real numbers. Select the combination(s) of values of the real parameters $\xi$ and $\eta$ such that $ f(x, y) = e^{(\xi x + \eta y)}$ is a solution of the given $\text{PDE}.$

  1. $\xi = \frac{1}{\sqrt{2a}}, \eta = \frac{1}{\sqrt{2b}}$
  2. $\xi = \frac{1}{\sqrt{a}}, \eta =0$
  3. $\xi = 0, \eta = 0$
  4. $\xi = \frac{1}{\sqrt{a}}, \eta = \frac{1}{\sqrt{b}}$
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