(a) Using the method of separation of variables, find the solution u(x, y) to the problem
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \quad (0 < x < 2, 0 < y < 1),$$
$$\frac{\partial u}{\partial x}(0, y) = \frac{\partial u}{\partial x}(2, y) = 0 \quad (0 < y < 1),$$
$$u(x, 1) = 0 \quad (0 < x < 2),$$
$$\frac{\partial u}{\partial y}(x, 0) = x \quad (0 < x < 2).$$
(b) Find the steady temperature distribution u(r, θ, φ) that satisfies Laplace's equation in the spherical region
$$r < R, \quad 0 \le \theta \le \pi, \quad -\pi < \phi \le \pi$$
(using spherical polar coordinates), when the temperature on the surface of the sphere is maintained at
$$u(R, \theta, \phi) = 3 \cos(\theta) \quad (0 \le \theta \le \pi, \quad -\pi < \phi \le \pi).$$
