Fuchs's theorem

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