Fuchs's theorem

In mathematics, the Fuchs's theorem states that 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

$y=\sum _{n=0}^{\infty }a_{n}(x-a)^{n+\lambda },\quad a_{0}\neq 0$

for some real $\lambda >0$, where its 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