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  "body": "\n## Summary\n\nResonance is the strong response of a linear system when it is forced at one of its own natural frequencies.\nFor a linear equation with constant coefficients and forcing $e^{\\lambda t}$, it happens when $\\lambda$ is a root of the characteristic polynomial.\nThen the response is not a multiple of $e^{\\lambda t}$ but picks up a power of $t$, and an undamped oscillator has unbounded solutions.\nIt explains the collapse of structures under periodic loads and the tuning of circuits and instruments.\nSee the [[definition|NTB5JQGPGC]] and the worked [[forced oscillator|0X4NT20S3G]].\n\n## Explanations\n\n### L1 human\n\nA child on a swing goes higher and higher when pushed in time with the swing.\nThe pushes add up because they match the swing's own rhythm.\nFor the undamped oscillator $x'' + x = \\cos t$, the forcing has the natural frequency $1$.\nA solution is $x(t) = \\frac{1}{2} t \\sin t$, whose swings grow without bound.\nWith forcing $\\cos(2t)$ instead, every solution stays bounded.\n",
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