Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 8C / b / Solution

For $m=1$, $Y=(1-{\eta }^2)/2$. Expanding both sides of the perturbation equation givesFor $y={\sum }_{r\ge 0}{a}_r{\eta }^r$, coefficient comparison givesThe even and odd series terminate exactly when $6\lambda =k(k+1)$ with the corresponding parity. Thus every positive integer $k$ gives the regular polynomial solution, a Legendre polynomial up to scale.