Question

Find the solution to Laplace’s equation which is bounded on all of $R^2$ such that on the circle . Write your answer in Cartesian coordinates.

u(r, y) = y y = 6

Answer 1

1. 
   Ans we Uxy)t 6 y)R form CIRペ bounded tor in ing u polar Y + pe Luth Consider R(r) De) Ulr,e) = RR R p(e) e) fi'rt talce p(e) ¢Ce) a0t6 here ab are Cenutamt we cam Conicler aetb = a Ttaetb ante) Poe)- a a-O tabe gecmdl K Eke (e) Ge tge No again P(o)dCet in) Ro
   
   4onlke)tb Sinlko ¢l0)= again Cel- uhere n,,3,4, de)-an Coa(nelt bSnne) K h d=-nt R"+O RCr) -CInlr)t finik fer C,O folT) ARK+YR'-Ra ut R RI=kK- R" KCK-)K put thue valus in above uat'en K(K-t) TK-3 + y krkt- Rale)= G again ) s frni at Rolr)= B, SinCne) rA Uy)ty
   
   U(r0)- YSine + +(G gne CfG,0)=G Sine CoA (a0)-1-9 Sin know U TG, e) 3 conle e) An CoACne)+ Sintne) Jr" Sin e- 3t 6 UCr,e) t nl UlE,) An Con (na) + Sin (ne) o t Am=0 Ro= 3; amd Aa = BI-I B=0 3t YSing (r,e)= now Sin Con 0= 4 NO vCX.y) = 3 ty-(X4) 3ty - (6-y3) +
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