Fuchs' theorem: Difference between revisions

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Given a second order differential equation of the form

y''+p(x)y'+q(x)y=g(x)

where $p(x)$ , $q(x)$ and $g(x)$ has power series expansions at $x=a$ . A solution to this second order differential equation can be expressed as a power series at $a$ . Thus any solution can be written as

y=\sum _{n=0}^{\infty }a_{n}(x-a)^{n}

,

where it's radius of convergence is at least as large as the minimum of the radii of convergence of $p(x)$ , $q(x)$ and $g(x)$ .

References

Asmar, Nakhlé H., "Partial differential equations with Fourier series and boundary value problems", ISBN: 0131480960

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 Given a second order differential equation of the form
-<math>y'' + p(x)y'+q(x)y=g(x)</math>
+:<math>y'' + p(x)y'+q(x)y=g(x)</math>
 where <math>p(x)</math>, <math>q(x)</math> and <math>g(x)</math> has power series expansions at <math>x=a</math>. A solution to this second order differential equation can be expressed as a power series at <math>a</math>. Thus any solution can be written as
-<math> y = \sum_{n=0}^\infty a_n (x-a)^n </math>,
+:<math> y = \sum_{n=0}^\infty a_n (x-a)^n </math>,
 where it's radius of convergence is at least as large as the minimum of the radii of convergence of <math>p(x)</math>, <math>q(x)</math> and <math>g(x)</math>.