= Solution
Set $\delta=\sqrt\epsilon$. The outer scale found below shows that the inner expansion requires the four asymptotic scales
$$
1,\qquad\delta,\qquad\epsilon\log\epsilon,\qquad\epsilon.
$$
Solving successively with $f_j(1)=0$ and matching the free homogeneous terms gives
$$
\boxed{
\begin{aligned}
f(r)\sim{}&
1-r^{-1/2}
+\sqrt{\pi\epsilon}\,(1-r^{-1/2})\\
&+\epsilon\log\epsilon\,(1-r^{-1/2})\\
&+\epsilon\left[
-\sqrt r+\log r+\pi+\frac C4
-\left(\pi+\frac C4-1\right)r^{-1/2}
\right].
\end{aligned}}
$$
This is valid for fixed $r=O(1)$.
To verify the differential-equation hierarchy, let $L[y]=y''+3y'/(2r)$. The leading terms satisfy
$$
L[f_0]=L[f_1]=0,
\qquad
f_0=1-r^{-1/2},
\qquad
f_1=\sqrt\pi f_0.
$$
At order $\epsilon$,
$$
L[f_2]=-f_0f_0',
$$
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.
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