In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form

$${\displaystyle y''+p(x)y'+q(x)y=g(x)}$$

${\displaystyle p(x)}$

${\displaystyle q(x)}$

${\displaystyle g(x)}$

${\displaystyle x=a}$

${\displaystyle a}$

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

$${\displaystyle y=y_{0}\ln(x-a)+\sum _{n=0}^{\infty }b_{n}(x-a)^{n+r},\quad b_{0}\neq 0}$$

Its radius of convergence is at least as large as the minimum of the radii of convergence of ${\displaystyle p(x)}$, ${\displaystyle q(x)}$ and ${\displaystyle g(x)}$.