1) Find the steady-state temperature in a sphere of radius R
(a) if the surface is held at $u(R, \cos \theta) = \cos^2 \theta$
(b) if the surface is held at
$u(R, \cos \theta) = \begin{cases} \cos \theta, & 0 < \theta < \pi/2 \text{ (top hemisphere)}\\ 0, & \pi/2 < \theta < \pi \text{ (bottom hemisphere)} \end{cases}$ (6p)