 Legendre differential equation

The equation $(1-x^2)y\prime \prime -2xy\prime +\ell (\ell +1)y=0$ has polynomial solutions for nonnegative integer $\ell$.

The equation $(1-x^2)y\prime \prime -2xy\prime +\ell (\ell +1)y=0$ has the Legendre polynomial of degree $\ell$ as its normalized regular polynomial solution when $\ell$ is a nonnegative integer.