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  "body": "\n## Statement\n\nLet $a_0, \\dots, a_{n-1}$ be real numbers, and let $p$ be the [[characteristic polynomial|0QDXMG46QW]] of the equation $y^{(n)} + a_{n-1} y^{(n-1)} + \\dots + a_0 y = 0$.\nLet $\\alpha + i \\beta$ be a root of $p$ of multiplicity $m$, with $\\alpha, \\beta \\in \\mathbb{R}$ and $\\beta \\ne 0$.\nThen $\\alpha - i \\beta$ is also a root of multiplicity $m$, and the $2m$ functions\n$$x^k e^{\\alpha x} \\cos(\\beta x), \\quad x^k e^{\\alpha x} \\sin(\\beta x), \\qquad k = 0, \\dots, m - 1,$$\nare linearly independent real solutions of the equation on $\\mathbb{R}$.\n\n## Remarks\n\nBy [[Euler's formula|04R3CY41VJ]], $e^{(\\alpha + i \\beta) x} = e^{\\alpha x} (\\cos \\beta x + i \\sin \\beta x)$. Its real and imaginary parts are solutions because the coefficients are real.\nFor order $2$ and roots $\\alpha \\pm i \\beta$, the real solutions are $e^{\\alpha x} (c_1 \\cos \\beta x + c_2 \\sin \\beta x)$.\nExample: $y'' + 2 y' + 5 y = 0$ has the roots $-1 \\pm 2 i$, so its solutions are $e^{-x} (c_1 \\cos 2x + c_2 \\sin 2x)$.\nThese families for all roots, together with those of [[real roots|0HRVY67CNV]], give a real fundamental system.\n\n## Explanations\n\n### L1 human\n\nComplex roots come in conjugate pairs, because the polynomial has real coefficients.\nEach pair gives an oscillation: a sine and a cosine with frequency $\\beta$.\nThe real part $\\alpha$ makes the oscillation grow if $\\alpha \u003e 0$ and decay if $\\alpha \u003c 0$.\n",
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