Laplace in Polar Coordinates: Solving the Polar Coordinate Equation

[dp182]

Homework Statement

solve the polar coord equation

Homework Equations

U_rr+(1/r)U_r+(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,$\vartheta$)=U_o(r,$\vartheta$)+$\sum$c_nR_n(r)$\vartheta$_n($\vartheta$)

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 R_n(r)=C_nr⁴ⁿ+d_nr^-4n and knowing condition U(1,$\vartheta$)=0 bn=0 so what's 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 I am missing
and then using the

 

[lippyka]

form U(r,\vartheta)=Uo(r,\vartheta)+\sumcnRn(r)\varthetan(\vartheta) I'm not sure where to go from here