(a) Legendre's differential equation may be written

$$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+n(n+1) y=0, \quad y(1)=1$$

Show that for non-negative integer $n$, this equation has a solution $P_{n}(x)$ that is a polynomial of degree $n$. Find $P_{0}, P_{1}$ and $P_{2}$ explicitly.

(b) Laplace's equation in spherical coordinates for an axisymmetric function $U(r, \theta)$ (i.e. no $\phi$ dependence) is given by

$$\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial U}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial U}{\partial \theta}\right)=0$$

Use separation of variables to find the general solution for $U(r, \theta)$.

Find the solution $U(r, \theta)$ that satisfies the boundary conditions

$$\begin{aligned} &U(r, \theta) \rightarrow v_{0} r \cos \theta \quad \text { as } r \rightarrow \infty \\ &\frac{\partial U}{\partial r}=0 \quad \text { at } r=r_{0} \end{aligned}$$

where $v_{0}$ and $r_{0}$ are constants.