Consider the 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 combinations of real parameters $\xi$ and $\eta$ such that
\[
f(x,y)=e^{\,(\xi x+\eta y)}
\]
is a solution of the PDE.