Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 31E / i / Solution

After division by $x^2$, the equation becomesThe regular singular point criterion for a second-order equation would require $x(1/x)$ and $x^2(-1/x^4)$ to be analytic at zero. The second expression is $-x^{-2}$, so the criterion fails: