2d potential problem for a region bounded by 4 planes

[samee]

Homework Statement

Consider a 2d potential problem for a region bound by 4 planes
x=-1/2
x=1/2
y=0
y=1

There are no charges inside the bounded region. The boundaries at y=0 and y=1 are held at zero potential. The potential at the boundaries x=-1/2 and x=1/2 is given by,
V(-1/2,y)=V(1/2,y)=Vosin(2piy)

(a) Find the electrostatic potential V(x,y) everywhere inside this region by solving the Laplace equation in 2 dimensions using the method of separation of variables.
(b) Calculate the surface charge density on the boundary y=0

Homework Equations

V(-1/2,y)=V(1/2,y)=Vosin(2piy)

The Attempt at a Solution

Okay, so I know that the potential is an integral, but it seems unsolvable to me! Help me at least set up this problem so I can go from there?

 

[betel]

The question explicitly mentions Laplace's equation. How does it look like?
If you have it, try solving it using the hint in the question.
The boundary values will enable you to write the potential everywhere.

 

[samee]

so,
dVosin(2piy)² = 0
d²x

dVosin(2piy)² = 2piVocos(2piy)
d²y

dVosin(2piy)² = 0
d²z

So then the laplace equation is 0+2piVocos(2piy)+0=2piVocos(2piy)

But this isn't an integral so it can't be V, can it?

 

[betel]

No, this only looks similar to Laplace's equation.
It will be a differential equation relation the potential to the charge density (zero here). Every differential equation needs some initial or boundary values. Here the boundary values are given by the value of the potential on the four planes.
Why don't you write down the general Laplace equation as you learned it in the course?