Power series solution

Summary

The power series method looks for a solution of a differential equation in the form $y = \sum_{k \ge 0} a_k (x - x_0)^k$. Consider $y'' + p(x) y' + q(x) y = 0$, where $p$ and $q$ are analytic at $x_0$: they equal convergent power series near $x_0$. One puts the series into the equation and compares the coefficients of each power $(x - x_0)^k$. This gives a recurrence for the $a_k$, with $a_0$ and $a_1$ free. The resulting series converge at least on each disc around $x_0$ where the series of $p$ and $q$ converge, and they give two independent solutions.