Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 31C / a / Solution

The coefficient changes sign where $1-{\lambda }^2/x^2=0$, so the single turning point isPut $p(x)=({\lambda }^2/x^2-1{)}^{1/2}$ and $q(x)=(1-{\lambda }^2/x^2{)}^{1/2}$. The WKB approximation satisfying the boundary condition at $x=1$ iswhile decay at infinity selects