Consider $\epsilon^2y''(x)=(\sin x)y$. For what fixed value of $x$ is the physical-optics approximation a good approximation? So I am wondering what is the general method for this kind of problem. From Bender's book, for equation in the form of $\epsilon^2y''=Q(x)y, Q(x)\neq 0$, we have a computed the $S_0,S_1, $and so on . The physical-optics approximation is just using the first two terms in the WKB. So do we need to require that $S_{n+1}/S_n$ for every $n$ to be bounded or is it sufficient to let $\delta S_2\ll 1$?
And for this specific question, how are we gonna determine the $x$'s? And is the WKB accurate as $x\to\infty?$ Any hints would be appreciated! Thanks!