(b) (5 marks) In the polar coordinates
$r := \sqrt{x^2 + y^2}$ and $\theta := \arctan(\frac{y}{x})$,
Laplace's equation on the unit disk for the unknown function $u = u(r, \theta)$ takes the
form
$\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0$, $0 < r < 1$, $0 \le \theta < 2\pi$.
Find the particular solution that satisfies the boundary condition
$u(1, \theta) = \cos \theta$, $0 \le \theta < 2\pi$.