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

With $X=ϵx$, the equation isThe leading WKB approximation for a slowly varying oscillator isIt requires smooth nonzero $k$, $ϵ|k^{\prime }|/|k{|}^2\ll 1$, and distance from every classical turning point large compared with its turning-point scale.

For $k(X)=\sqrt{1+X^2}$,Since $k(0)=1$ and $k^{\prime }(0)=0$, the initial data select the cosine branch without an $O(1)$ phase correction. Hence, for $x>0$ in the WKB regime,

The reduced first-order equation cannot satisfy both endpoint values. The given outer expansion satisfies $y(1)=2$ but has $y_0(0)=e$, so the boundary layer lies at $x=0$ and has stretched coordinate $X=x/ϵ$.

Write the inner expansion as $Y=Y_0+ϵY_1+\cdots$. After multiplying the differential equation by $ϵ$, it becomesAt leading order,The boundary condition $Y_0(0)=0$ and matching to $y_0(0)=e$ giveAt the next order,Matching to $y_1(0)=e(\ln 2-1)$ and imposing $Y_1(0)=0$ gives

The additive composite expansion is outer plus inner minus their common part. With $X=x/ϵ$, it isIt satisfies $y_{\mathrm{comp}}(0)=0$ exactly through the retained order, satisfies the right boundary condition up to exponentially small terms, and is uniformly accurate to $O(ϵ)$.