Laplace’s equation in planar polar co-ordinates (r, µ) is

Frr + 1/r Fr + 1/r^2 Fµµ = 0.

Show, for any integer k, that

F (r, µ) = r^k cos (kµ) and F (r, µ) = r^k sin (kµ)

are solutions to Laplace’s equation.

(i) Find a bounded solution to Laplace’s equation in the region r <= 1 which satisfies the boundary condition F (1, µ) = cos^2 µ for 0 <= µ < 2pi.

(ii) Find a bounded solution to Laplace’s equation in the region r >= 2 which satisfies the boundary condition F (2, µ) = sin^2 µ for 0 <= µ < 2pi

I can do the first part, showing that they are solutions to Laplace's equation. I'm totally stuck as to how to do i) and ii).. is this to do with Dirichlet's Problem and seperable solutions? Thanks!