Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 2 / a / i / Solution

Differentiate the power series term by term inside its radius of convergence, and reindex the second derivative so that every coefficient multiplies $x^m$. The differential equation becomesThe uniqueness of power-series coefficients givesThe even coefficients depend only on $a_0$ and the odd coefficients only on $a_1$. These are the two local solution branches of the Legendre differential equation around its ordinary point $x=0$.