# Regular singular points of 2nd order ODE

1. May 27, 2010

### Jerbearrrrrr

1. The problem statement, all variables and given/known data
[PLAIN]http://img265.imageshack.us/img265/6778/complex.png [Broken]

I did the coefficient of the w' term. What about the w term?

This seems like a fairly standard thing, but I can't seem to find it anywhere.
What ansatz should I use for q, if the eqn is written w''+pw'+qw?
C/(z-a)²+ D/(z-b)²?
Any conditions, except for the one generated by z->1/t substitution?
Or should I use C+cz on the top, etc?

Last edited by a moderator: May 4, 2017
2. May 27, 2010

### gabbagabbahey

Well, if a 2nd order ODE has regular singular points at $z=a$ and $z=b$, then $q$ has poles up to 2nd order at those points, and the most general form of $q$ is then

$$q(x)=\frac{g(z)}{(z-a)^2(z-b)^2}$$

where $g(z)$ is analytic everywhere. You should have used similar reasoning to find

[tex]p(z)=\frac{f(z)}{(z-a)(z-b)}[/itex]

Then just apply the linearity condition to find $f$ and $g$.

Last edited by a moderator: May 4, 2017