= Nested layers at an essential singular endpoint
{title2=$\epsilon y\prime\prime+(1+2\epsilon/x-2\epsilon^3/x^2)y\prime+2y/x=0$}
Multiplication by $x^2$ makes this example exactly integrable once: $\epsilon x^2y\prime+(x^2-2\epsilon^3)y=C$. Its homogeneous solution is $\exp(-x/\epsilon-2\epsilon^2/x)$. A solution with $y(1)=\epsilon^3$ that remains uniformly bounded needs $C\sim\epsilon^3$, hence $y(0)\sim-1/2$. The scales $x=O(\epsilon)$ and $x=O(\epsilon^2)$ connect the small outer solution to the order-one endpoint value. Bounded homogeneous contributions remain invisible to algebraic matching at the right boundary.
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