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  "body": "\n## Definition\n\nLet $a_0, \\dots, a_{n-1}$ be real numbers, and consider the homogeneous [[linear equation with constant coefficients|0A6PGTTP2A]]\n$$y^{(n)} + a_{n-1}\\, y^{(n-1)} + \\dots + a_1\\, y' + a_0\\, y = 0.$$\nIts characteristic polynomial is $p(r) = r^n + a_{n-1} r^{n-1} + \\dots + a_1 r + a_0$.\nIts characteristic equation is the polynomial equation $p(r) = 0$ for an unknown complex number $r$.\n\n## Remarks\n\nFor each complex $r$, the function $y = e^{r x}$ satisfies $y^{(n)} + \\dots + a_0 y = p(r)\\, e^{r x}$. So $e^{rx}$ is a solution if and only if $p(r) = 0$.\nExample: the equation $y'' - 3 y' + 2 y = 0$ has the characteristic equation $r^2 - 3 r + 2 = 0$, with roots $1$ and $2$.\nThe [[characteristic polynomial of a linear recurrence|0BWDJGQYST]] plays the same role for recurrences, with $r^k$ in place of $e^{r x}$.\n\n## Explanations\n\n### L1 human\n\nTry a solution of the form $e^{r x}$.\nEach derivative multiplies it by $r$, so the equation becomes $p(r) e^{rx} = 0$.\nSince $e^{rx}$ is never zero, $r$ must be a root of $p$.\n",
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