Can this method be applied to any ODE with a regular singular point at $x=0$?

[oasi]

How can we solve this ODE with series

http://img841.imageshack.us/img841/8682/80858005.png

 

[Dustinsfl]

How can we solve this ODE with series

http://img841.imageshack.us/img841/8682/80858005.png

Start by letting $y=\sum\limits_{n=0}^{\infty}c_nx^n$ and taking the appropriate derivatives and plugging them into the DE.

 

[Ackbach]

How can we solve this ODE with series

http://img841.imageshack.us/img841/8682/80858005.png

Start by letting $y=\sum\limits_{n=0}^{\infty}c_nx^n$ and taking the appropriate derivatives and plugging them into the DE.

Actually, since $x=0$ is a regular singular point of this DE, you're going to have to use the Method of Frobenius. Try
$$y=\sum_{n=0}^{\infty}c_{n}x^{n+r}.$$