As we know, if the equation
$$a(x)y''+b(x)y'+c(x)=0 \ \ \ \ \ \ \ \ \ (1)$$


has a regular singular point at $x=x\_0$ then we seek solution of the equation as 


$$y(x)=\sum\_{n=0}^{\infty}\beta\_n (x-x\_0)^{n+\lambda} \ \ \ \ \ \ \ (2)$$


but what about the case when the equation $(1)$ has several singular points? At which point should I expand (2) series ? What principle should I use to take right expansion point?



Frobenius method for multiple singular points
You may expand at every point, singular or non-singular. At a non-singular point
you will obtain 2 series with integer powers, with radius of convergence at least the distance to the closest singular point. At a regular singular point you obtain two series with radius of convergence at least the distance to the other singular points. You can do it for every singular (and non-singular) point, and obtain many series, each representing solutions in its own disk. These discs overlap, of course. The problem how Frobenius solutions on two different overlapping disks are related
is a difficult problem which is called the "connection problem".


In which point you "should" expand depends on the purpose of your expansion.


The standard reference is E. L. Ince, Ordinary differential equations.


