Verify that the function uxyalnx2y2b satisfies Laplaces equa

Verify that the function u(x,y)=aln(x^2+y^2)+b satisfies Laplace\'s equation and determine a and b so that u satisfies the boundary conditions u=110 on the circle x^2+y^2=1 and u=0 on the circle x^2+y^2=80.

Solution

u(x,y) = aln(x²+y²)+b

u_x = 2ax/(x²+y²)    u_xx = {2a(x²+y²)-2ax(2x)}/(x²+y²)²

u_y = 2ay/(x²+y²) ;    u_yy ={2a(x²+y²)-2ay(2y)}/(x²+y²)²

u_xx + u_{yy =} 4a/(x²+y²) - 4ax²/(x2+y2)-4ay²/(x2+y2)

= 4a/(x²+y²) - 4a/(x²+y²)=0

Thus u satisfies Laplace equation.

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Boundary conditions are u =110, when (x²+y²)=1

i.e. 110 = aln1 +b

b =110

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u =0 on circle (x²+y²)=80

0 = a ln 80+110

a ln 80 = -110

a = -110/ln 80