Answer
The Laplace's equation is satisfied.
Work Step by Step
We need to verify Laplace's equation
$\nabla^2 f =\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}=0$
We take the first partial derivative of the given function $f(x,y)$ with respect to $x$, by treating $y$ as a constant, and vice versa:
$f_x=-2 e^{-2y} \sin 2x \\ f_y=-2 e^{-2y} \cos 2x $
Now, $\nabla^2 f =-2 e^{-2y} \sin 2x-2 e^{-2y} \cos 2x +4 e^{-2y} \cos 2x=0$
So, the Laplace's equation is satisfied.