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

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.

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Introduce the stretched coordinateThe equation becomesSeek the three-term outer expansionAt first order,so decay at infinity and matching giveAt the next order,Using the function $Y$ supplied in the question, the matched decaying solution isHere $Y$ is the stated particular solution normalized to decay at infinity.

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Use $\rho =ϵr={\delta }^{2}r$ in the supplied small-$\rho$ expansions. The outer approximation re-expanded in inner variables isThis is exactly the large-$r$ re-expansion of the inner result in part 1. In particular, the identityforces the switchback term $ϵ\log ϵ(1-r^{-1/2})$. The coefficient of the homogeneous $E_{3/2}$ contribution at outer order $ϵ$ is fixed to $-2\sqrt{\pi }$, while the remaining inner homogeneous constant is fixed to $\pi +C/4-1$. Thus the two matched asymptotic expansions agree term by term in $1\ll r\ll {ϵ}^{-1}$.

Solved by gpt-5.6-sol high.