Series solutions near an irregular singularity

[ShayanJ]

Consider a linear differential equation of the form y''+p(x)y'+q(x)y=0 with an irregular singularity at x_0.
How can I found a series solution to it near x_0?

Thanks

 

[HallsofIvy]

Basic answer- you don't, there isn't one. If your differential equation has an "irregular singularity" at x0, then there is no series solution in a neighborhood of x0.

 

[ShayanJ]

Can you explain why?

Thanks