Consider Legendre's equation

$$\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\lambda y=0 .$$

Show that if $\lambda=n(n+1)$, with $n$ a non-negative integer, this equation has a solution $y=P_{n}(x)$, a polynomial of degree $n$. Find $P_{0}, P_{1}$ and $P_{2}$ explicitly, subject to the condition $P_{n}(1)=1$.

The general solution of Laplace's equation $\nabla^{2} \psi=0$ in spherical polar coordinates, in the axisymmetric case, has the form

$$\psi(r, \theta)=\sum_{n=0}^{\infty}\left(A_{n} r^{n}+B_{n} r^{-(n+1)}\right) P_{n}(\cos \theta)$$

Hence, find the solution of Laplace's equation in the region $a \leqslant r \leqslant b$ satisfying the boundary conditions

$$\begin{cases}\psi(r, \theta)=1, & r=a \\ \psi(r, \theta)=3 \cos ^{2} \theta, & r=b\end{cases}$$