0
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ You should add one of the top level domain tags for better visibility. $\endgroup$ Oct 29, 2015 at 14:30

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Thank you very much for your interesting answer. Could you please tell me where can i find more information about Frobenius method and "connection problem"? $\endgroup$
    – Paramore
    Oct 29, 2015 at 20:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.