Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 8 / c / Solution

Normalize the free leading coefficient to $a_0=1$. The linear recurrence relation from part (b) givesConsequently the first four nonzero coefficients areand the single pure Frobenius solution isFor clarity about the second independent solution, a choice with no added multiple of $y_1$ isThese companion coefficients follow by reduction of order with the normalized Wronskian $W=z^{-3}(1+z^2{)}^{3/2}\exp [3z^2/(2(1+z^2))]$. The logarithm requires a local branch; it does not supply a second ordinary power series solution.