Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 8 / c

For $\ell =0$ the terminating even solution is constant; for $\ell =1$ the terminating odd solution is proportional to $x$. For $\ell =2$, the recurrence gives $a_2=-3a_0$; for $\ell =3$ it gives $a_3=-5a_1/3$. Imposing the Legendre polynomial normalization at $1$ yields