Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2008 / A / 2 / i / Solution

Write $y=e^{S/\delta }$ and divide the equation by $y$:Balancing the first and third terms gives $\delta =ϵ$. With $S=S_0+ϵS_1+\cdots$, the terms of order ${ϵ}^{-1}$ and one giveFor the slow branch $S_0^{\prime }=0$, choose $S_0=0$ and obtain $S_1=-x^2/2$. For the fast branch $S_0^{\prime }=-1$, choose $S_0=-x$ and obtain $S_1=x^2/2$. Additive integration constants enter the two independent amplitudes. The two-term WKB approximation is consequentlyImposing the boundary conditions on these two approximating modes yieldsThis uses the first two exponent functions $S_0,S_1$ for both branches. The next transport equations, if desired to identify the algebraic error, give $S_2=x-x^3/3$ for the slow mode and $S_2=x+x^3/3$ for the fast mode. Thus each displayed mode has relative amplitude correction $1+O(ϵ)$ uniformly on the fixed interval; no turning point occurs because the two leading exponent roots remain distinct.