Does an Electric Field Exponentially Decrease in Free Space?

[win_lan]

Homework Statement

Can an electric field drops exponentially? (in free space)

Homework Equations

Starting from a hypothetical potential, $$V = F(x,y,z)e^{-y^2}$$ which decays exponentially in the y direction
$$\nabla^2V=0$$

The Attempt at a Solution

$$\nabla^2V=e^{-y^2}(\frac{\partial^2F}{\partial x^2}+\frac{\partial^2F}{\partial z^2}+\frac{\partial^2F}{\partial y^2}-4y\frac{\partial F}{\partial y}+(4y^2-2)F)=0$$
Using separable variable, the general solution of $$\frac{\partial^2F}{\partial y^2}-4y\frac{\partial F}{\partial y}+(4y^2-2)F=0$$ has the form $$(A+By)e^{y^2}$$, assuming that all general solutions of the above equation can be expressed as a linear combination of the product of the 3 individual solution of the separable variable, we can see that the exponential term will cancel out.

I am not sure how to proceed from here, is this correct? does it mean we cannot have the exponential term in either potential or e field?

 

[Rainbow Child]

You forgot the separation constant, i.e.

$$\frac{\partial^2F}{\partial y^2}-4y\frac{\partial F}{\partial y}+(4y^2-2)F=k^2$$