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  "body": "\n## Definition\n\nLet $p(r) = r^n + a_{n-1} r^{n-1} + \\cdots + a_0$ be a polynomial with complex coefficients, and write $D = \\frac{d}{dt}$.\nConsider the [[linear equation|0QQRP17Z7S]] with constant coefficients and exponential forcing\n$$p(D)\\, y = e^{\\lambda t}, \\qquad \\lambda \\in \\C.$$\nThe equation is *resonant*, and $\\lambda$ is a *resonant frequency*, if $p(\\lambda) = 0$, that is, if $\\lambda$ is a root of the [[characteristic equation|0GZWXM2DKE]] of $p(D) y = 0$.\n\n## Remarks\n\nIf $p(\\lambda) \\ne 0$, then $y = \\frac{e^{\\lambda t}}{p(\\lambda)}$ is a particular solution.\nIf $\\lambda$ is a root of multiplicity $m \\ge 1$, then $y = \\frac{t^m e^{\\lambda t}}{p^{(m)}(\\lambda)}$ is a particular solution.\nThis follows from $p(D)(t^m e^{\\lambda t}) = \\frac{\\partial^m}{\\partial \\lambda^m} \\left( p(\\lambda) e^{\\lambda t} \\right)$ and $p(\\lambda) = \\cdots = p^{(m-1)}(\\lambda) = 0$.\n\nExample: $y'' + \\omega_0^2 y = e^{i \\omega_0 t}$ with $\\omega_0 \u003e 0$ is resonant, since $p(r) = r^2 + \\omega_0^2$ vanishes at $i \\omega_0$.\nTaking real parts, $y'' + \\omega_0^2 y = \\cos(\\omega_0 t)$ has the solution $y = \\frac{t \\sin(\\omega_0 t)}{2 \\omega_0}$.\nIndeed $y'' = \\cos(\\omega_0 t) - \\frac{\\omega_0 t \\sin(\\omega_0 t)}{2}$, so $y'' + \\omega_0^2 y = \\cos(\\omega_0 t)$; this solution is unbounded.\n\nNon-example: $y'' + y = e^{2 i t}$ is not resonant, since $p(2i) = -3 \\ne 0$; the particular solution $-\\frac{1}{3} e^{2 i t}$ is bounded.\n\nFor a damped oscillator, $p(r) = r^2 + c r + \\omega_0^2$ with $c \u003e 0$, the roots of $p$ have negative real part, so a forcing $e^{i \\omega t}$ with real $\\omega$ is never resonant in this sense.\nWhen the damping is small, the amplitude $\\frac{1}{\\abs{p(i \\omega)}}$ still has a sharp peak near the natural frequency; this is also called resonance.\n",
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