Suppose we have $\epsilon y''+\frac{1}{x}y'+y=0$. $y(1)=e^{-1/2}, y(0)$ finite,
Suppose we have shown there is no boundary layer at $x=0$ even though $\frac{1}{x}\geq 0$(caused by the regular singularity of $0$). And we consider the outer expansion $y_{out}\sim y_0(x)+\epsilon y_1(x)+......$ We need to show it is not a regular perturbation expansion, that is, we want to show it diverges.
Bender's book suggest that we can look at the exact solution to our problem, which is $$ y=e^{-1/2}x^{(\epsilon-1)/2\epsilon}\frac{J_{1/2-1/2\epsilon}(x/\sqrt{\epsilon})}{J_{1/2-1/2\epsilon}(1/\sqrt{\epsilon})} $$ How do we use this? Can anyone gives some hint please?
Thanks!