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I have just learned to solve ODEs via power series method. I am a bit confused about the Fuchs' theorem. The theorem mentions ordinary and regular singular points at $x_0$ and our ability to find the power series solution. What is $x_0$? Is it any point on the domain of the ODEs solution or is it a specific point? Is a power series (Frobenius) method based on fitting a series around the point or something else?

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    $\begingroup$ This question does not seem to have physical content. Perhaps Mathematics would be a better place to ask. $\endgroup$ – Danu Sep 24 '14 at 11:28
  • $\begingroup$ OK, let's close this question. $\endgroup$ – V. Moretti Sep 24 '14 at 13:58
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Your series expansion is taken about the point $x_{0}$, which may be an ordinary, regular singular, or irregular singular point.

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  • $\begingroup$ But what is this $x_0$ point? Is it any point on the interval of the function? Is there any similarity with $x_0$ and $x=a$ in Taylor series, where we are building a series around a certain point? $\endgroup$ – Mark Oct 1 '14 at 16:09
  • $\begingroup$ It's exactly like in Taylor Series. If someone came up to you in the street with a differential equation and said "Find a series solution" then theu would need to give you a point about which to expand that series. Sometimes the solution at a particular point won't be very well defined - these are the singular points. $\endgroup$ – preferred_anon Oct 1 '14 at 16:28

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