With $X=\epsilon x$, the equation is
$$
\epsilon^2y_{XX}+k^2(X)y=0.
$$
The leading <WKB approximation for a slowly varying oscillator> is
$$
\boxed{
y\sim\frac1{\sqrt{k(X)}}\left[
C_+e^{iS(X)/\epsilon}+C_-e^{-iS(X)/\epsilon}
\right],
\qquad
S(X)=\int^Xk(s)ds.}
$$
It requires smooth nonzero $k$, $\epsilon|k'|/|k|^2\ll1$, and distance from every <classical turning point> large compared with its turning-point scale.

For $k(X)=\sqrt{1+X^2}$,
$$
S(X)=\int_0^X\sqrt{1+s^2}ds
=\frac12\left[X\sqrt{1+X^2}+\operatorname{arsinh}X\right].
$$
Since $k(0)=1$ and $k'(0)=0$, the initial data select the cosine branch without an $O(1)$ phase correction. Hence, for $x>0$ in the WKB regime,
$$
\boxed{
y(x)\sim(1+\epsilon^2x^2)^{-1/4}
\cos\left\{
\frac{\epsilon x\sqrt{1+\epsilon^2x^2}
+\operatorname{arsinh}(\epsilon x)}{2\epsilon}
\right\}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 336 / 3 / a / Solution

For $\epsilon^2y_{XX}+k^2(X)y=0$ with nonzero slowly varying $k$,
$$
y\sim k(X)^{-1/2}\left[
C_+e^{i\epsilon^{-1}\int^Xk(s)ds}
+C_-e^{-i\epsilon^{-1}\int^Xk(s)ds}
\right].
$$
It fails near turning points and where $\epsilon|k'|/|k|^2$ is not small.


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