laplace in polar coords

1. The problem statement, all variables and given/known data
solve the polar coord equation

2. Relevant equations
Urr+(1/r)Ur+(1/r2)U$\vartheta$$\vartheta$=0; 1$\leq$r$\leq$2, 0$\leq$$\vartheta$$\leq$$\pi$/4
U(1,$\vartheta$)=0 U(2,$\vartheta$)=$\vartheta$
U$\vartheta$(r,0)=0 U($\vartheta$,$\pi$/4)=0
and has the form
U(r,$\vartheta$)=Uo(r,$\vartheta$)+$\sum$cnRn(r)$\vartheta$n($\vartheta$)

3. The attempt at a solution
so I used variation of parameters and the homogeneous boundaries with $\vartheta$ to get $\vartheta$n($\vartheta$)=cos(4n$\vartheta$) then taking r2R''+rR'-cR=0 and using the Euler equation to get r(r-1)+r-c which is r=$\pm$c, c=16n2
so that will give me Rn(r)=Cnr4n+dnr-4n and knowing condition U(1,$\vartheta$)=0 bn=0 so whats left is Cnr4n I got stuck here how do I solve for Cn do I try subbing in conditions to solve or is there some equation im missing
and then using the
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 Tags laplace transform, pde, polar coordinates