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Consider the singular ODE

$y''+\frac{y'}{r}+p(r)y=0 \ \ with \ \ y(0)=1 \ \ and \ \ y'(0)=0$.

Theoretically such solution exists and is unique if $p$ is nice. Is there a method to numerically solve this equation on some interval $[0, \alpha]$ with an explicit global error bound? I am not sure how to impose the initial condition at the singular point $r=0$.

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The SLEIGN2 software package can handle Sturm-Liouville problems with regular as well as singular end points. I believe you can find some papers on that web page that document various aspects of the methods used in the code.

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