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  "body": "\n## Summary\n\nMathieu functions are solutions of Mathieu's equation $y'' + (a - 2q \\cos 2x) y = 0$.\nFor each $q$, only special values of $a$, the characteristic values, admit solutions of period $\\pi$ or $2\\pi$.\nThese periodic solutions $\\operatorname{ce}_r(x, q)$ and $\\operatorname{se}_r(x, q)$ are the Mathieu functions in the narrow sense.\nThey arise when the [[Helmholtz equation|MNXKTXJ35G]] is separated in elliptic coordinates, for instance for vibrations of an elliptic drum.\nMathieu's equation is the basic example of a [[Hill differential equation|YE7H8DGDPY]], studied by [[Floquet theory|BJVZXPB297]].\n\n## Explanations\n\n### L1 human\n\nMathieu's equation describes an oscillator whose stiffness is shaken periodically in time.\nFor $q = 0$ the shaking is off, and the equation $y'' + a y = 0$ has the periodic solutions $\\cos rx$ and $\\sin rx$ when $a = r^2$.\nFor $q \\ne 0$ periodic solutions still exist, but only when $a$ is tuned to special values $a_r(q)$ or $b_r(q)$ that move with $q$.\nThe periodic solutions are deformed cosines $\\operatorname{ce}_r$ and deformed sines $\\operatorname{se}_r$.\nFor other values of $a$ the solutions are either bounded but not periodic, or they grow exponentially, which is the phenomenon of parametric resonance.\n",
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