The Legendre equation is

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

for $-1 \leqslant x \leqslant 1$ and non-negative integers $n$.

Write the Legendre equation as an eigenvalue equation for an operator $L$ in SturmLiouville form. Show that $L$ is self-adjoint and find the orthogonality relation between the eigenfunctions.