This might be a topic for the [Strugges with the Continuum](https://johncarlosbaez.wordpress.com/2016/09/23/struggles-with-the-continuum-part-7/) blog post
Consider the equation that I derived for the [QBO time-series](http://contextearth.com/compact-qbo-derivation/)
If you look closely at the DiffEq, there is a clear singularity in the Sturm-Liouville equation. Its most obvious in the secant. When the arg reaches $\pi/2$ it hits the singularity. Yet the solution is bounded. What this means is that the solution either exactly zero-compensates for the infinity or its too narrow a delta to be important.
If you do a numerical integration of this DiffEq, it can get tricky with regards to picking the right delta interval, yet the closed-form solution is trivial.