Solution
= Solution
The <Liouville-Green exponential ansatz>[Liouville-Green approximation] gives
$$
u_±\sim q^{-1/4}\exp\left(±i\int^x\sqrt q\,ds\right).
$$
Here
$$
q^{-1/4}=x^{1/4}(1+1/(4x)+5/(32x^2)+\cdots),
$$
$$
\int^x\sqrt q\,ds=2\sqrt x+x^{-1/2}+x^{-3/2}/12+\cdots.
$$
The control ratio $|q\prime|q^{-3/2}=O(x^{-1/2})$ tends to zero, verifying consistency.