Is it possible to have a second order differential equation with multiple regular singular points? All of the examples I've encountered thus far only give one regular singular point or one regular singular point and one irregular singular point. Can someone provide an example of an equation with more than one regular singular point and show how to find the limit at the second singular point?
For example, $$x (x-1) y'' + y = 0$$ has regular singular points at $x=0$ and $x=1$. I presume you know how to solve it around $x=0$. To solve around $x=1$, just do the change of variables $t = x-1$: $$ (t+1) t y'' + y = 0$$ and solve this around $t=0$.
Hmm: actually, in this case you might want to try $t = 1-x$, as then the equation is equivalent to the original one. That is, if $y = Y(x)$ is a solution of the equation, then so is $y = Y(1-x)$.