Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 5C / a / Solution

For a normalized equation$x_0$ is an ordinary point when $P,Q$ are analytic there. It is a regular singular point when $(x-x_0)P(x)$ and $(x-x_0{)}^{2}Q(x)$ are analytic there. These are the ordinary point criterion for a second-order equation and regular singular point criterion for a second-order equation.

Hereso $x=0$ is regular singular. Seek a power-series solution of a differential equationEquating the coefficient of $x^n$ givesand henceThereforeThe series terminates exactly when $\lambda =N$ is a nonnegative integer: the factor with $j=N$ then makes $a_{N+1}=0$. Thus the polynomial solutions occur for