Fuchs's theorem

Given a second order differential equation of the form

$y"\; +\; p(x)y\text{'}+q(x)y=g(x)$

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