laplace in polar coords

dp182

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


2. Relevant equations
U_rr+(1/r)U_r+(1/r²)U_{[itex]\vartheta[/itex][itex]\vartheta[/itex]}=0; 1[itex]\leq[/itex]r[itex]\leq[/itex]2, 0[itex]\leq[/itex][itex]\vartheta[/itex][itex]\leq[/itex][itex]\pi[/itex]/4
U(1,[itex]\vartheta[/itex])=0 U(2,[itex]\vartheta[/itex])=[itex]\vartheta[/itex]
U_{[itex]\vartheta[/itex]}(r,0)=0 U([itex]\vartheta[/itex],[itex]\pi[/itex]/4)=0
and has the form
U(r,[itex]\vartheta[/itex])=U_o(r,[itex]\vartheta[/itex])+[itex]\sum[/itex]c_nR_n(r)[itex]\vartheta[/itex]_n([itex]\vartheta[/itex])

3. The attempt at a solution
so I used variation of parameters and the homogeneous boundaries with [itex]\vartheta[/itex] to get [itex]\vartheta[/itex]_n([itex]\vartheta[/itex])=cos(4n[itex]\vartheta[/itex]) then taking r²R''+rR'-cR=0 and using the Euler equation to get r(r-1)+r-c which is r=[itex]\pm[/itex]c, c=16n²
so that will give me R_n(r)=C_nr⁴ⁿ+d_nr^-4n and knowing condition U(1,[itex]\vartheta[/itex])=0 bn=0 so whats left is C_nr⁴ⁿ I got stuck here how do I solve for C_n do I try subbing in conditions to solve or is there some equation im missing
and then using the