Legendre's equation may be written

$$\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+n(n+1) y=0 \quad \text { with } \quad y(1)=1$$

Show that if $n$ is a positive integer, this equation has a solution $y=P_{n}(x)$ that is a polynomial of degree $n$. Find $P_{0}, P_{1}$ and $P_{2}$ explicitly.

Write down a general separable solution of Laplace's equation, $\nabla^{2} \phi=0$, in spherical polar coordinates $(r, \theta)$. (A derivation of this result is not required.)

Hence or otherwise find $\phi$ when

$$\nabla^{2} \phi=0, \quad a<r<b$$

with $\phi=\sin ^{2} \theta$ both when $r=a$ and when $r=b$.