How to solve the Laplace equation in a half disk with given boundary conditions?

[EsponV]

Homework Statement

Solve the Laplace equation: delta u = d²u/dx²+d²u/dy²
inside the half disk 0<r<R, 0<phi<pi
Temperature on the bottom side of the disk is zero, u(x,y=0)=0. Temperature on the upper side of the disk is u(r=R, theta) = u₀(phi), 0<phi<pi

Homework Equations

I'm assuming the Laplace equation and the given bc conditions

The Attempt at a Solution

I attempted the solution, which I think up until this point is correct, but I've gotten stuck trying to figure out the constants.

http://img708.imageshack.us/img708/707/p2150074.jpg

Thank you all for your help.

 

[EsponV]

I just got a email telling me I had a response, however I don't see it in the thread. Anyhow, the picture that only shows half the page of paper because that's all the work I had done on it. I got stuck at that point and wasn't sure how to go about calculating the unknown constants (if I had done everything correctly to that point).