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  "body": "\n## Summary\n\nThe power series method looks for a solution of a differential equation in the form $y = \\sum_{k \\ge 0} a_k (x - x_0)^k$.\nConsider $y'' + p(x) y' + q(x) y = 0$, where $p$ and $q$ are analytic at $x_0$: they equal convergent power series near $x_0$.\nOne 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.\nThe 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.\n\n## Remarks\n\nExample: for $y' = y$, the series $\\sum a_k x^k$ gives $(k + 1) a_{k+1} = a_k$, so $a_k = a_0 / k!$ and $y = a_0 e^x$.\nExample: for $y'' + y = 0$, the recurrence is $a_{k+2} = -a_k / \\left((k+1)(k+2)\\right)$. It gives the series of $\\cos x$ and $\\sin x$.\nAt a point where a leading coefficient vanishes, the Frobenius method, a variant with a factor $(x - x_0)^r$, is used.\n\n## Explanations\n\n### L1 human\n\nWrite the unknown function as a power series with unknown coefficients.\nThe equation turns into a rule that gives each coefficient from the earlier ones.\nSo you can compute the solution term by term, even when there is no closed formula.\n",
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