Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 2 / b

Set $X=ϵx$. The equation becomesSince $k$ is positive for every $X\ge 0$, there is no classical turning point. The leading WKB approximation for a slowly varying oscillator isThe condition $y(0)=0$ selects a sine, and $y_x(0)=1$ fixes its coefficient. Sincethe result isThe WKB validity measureis uniformly $O(ϵ)$ and tends to zero at both ends of $X\ge 0$. With no turning point, the approximation is therefore uniformly valid on the entire stated half-line.