The equation is Uxx + Uyy = 0
And domain of solution is 0 < x < a, 0 < y < b
Boundary conditions:
Ux(0,y) = Ux(a,y) = 0
U(x,0) = 1
U(x,b) = 2
What I've done is that I did separation of variables:
U(x,y)=X(x)Y(y)
Plugging into the equation gives:
X''Y + XY'' = 0
Rearranging:
X''/X = -Y''/Y = k...