Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 31A (Asymptotic Methods)

After setting $z=1/x$, the transformed coefficients have a singularity beyond the regular-singular growth bounds at $z=0$. Thus infinity is an irregular singular point.

The standard removal of the first derivative, $y=u/x$, gives $u\prime \prime +q(x)u=0$ with $q(x)=x^{-1}-x^{-2}$.

The Liouville-Green approximation givesHereThe control ratio $|q\prime |q^{-3/2}=O(x^{-1/2})$ tends to zero, verifying consistency.

Since $y=u/x$, two real independent approximations are