Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 336 / 2 / 1

Set $\delta =\sqrt{ϵ}$. The outer scale found below shows that the inner expansion requires the four asymptotic scalesSolving successively with $f_j(1)=0$ and matching the free homogeneous terms givesThis is valid for fixed $r=O(1)$.

To verify the differential-equation hierarchy, let $L[y]=y^{\prime \prime }+3y^{\prime }/(2r)$. The leading terms satisfyAt order $ϵ$,whose particular integral is $-\sqrt{r}+\log r+1$; the displayed homogeneous multiple of $1-r^{-1/2}$ is fixed by matching.

Solved by gpt-5.6-sol high.