Find the solutions to $\displaystyle \bigtriangledown^2 u = 0 $in two and three dimensions subject to the following boundary conditions

(a) $\displaystyle u(x) = u_0 $ for $\displaystyle ||x|| = a $ and $\displaystyle u = u_1 $ for $\displaystyle ||x|| = b$

(b) $\displaystyle u(x) = u_0 $ for $\displaystyle ||x|| = a $ and $\displaystyle \bigtriangledown u \cdot n = k $ for $\displaystyle ||x|| = b$

Note that the boundary conditions do not depend on $\displaystyle \theta $or $\displaystyle \phi $ (in 3D)