Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 336 / 3

The reduced outer equation is $(1+x)y_0^{\prime }+y_0=0$. Imposing the right boundary condition givesThe condition at $x=0$ requires $X=x/ϵ$. The first two inner terms areThey match $2+ϵ(-2X+3/2)$. The additive composite expansion is

On $[-1,1]$, the coefficient $1+x$ vanishes at the left endpoint. The ordinary $O(ϵ)$ exponential layer is replaced by a turning-point endpoint region of width $O(\sqrt{ϵ})$, where all three terms in the equation enter the leading balance.

The outer expansion behaves as $f\sim x^{-1}$ near zero. The terms $x$ and $ϵf$ become comparable when $x^2=O(ϵ)$, soThe equation has the exact first integralwhere the constant follows from $f(1)=2$. ThusWriting $F=F_0+ϵF_1+\cdots$ and choosing the branch matching the positive outer solution givesHenceIts large-$X$ expansion matches the supplied outer series.