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

Since $y=x^{-1/2}u$, the two linearly independent solutions of the original equation have the asymptotic expansionsAs a check, the substitution $z=1/x$ converts the equation into the modified Bessel equation of order zero, whose solutions $I_0(z)$ and $K_0(z)$ have exactly these large-$z$ expansions.

Solved by gpt-5.6-sol high.