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  "body": "\n## Summary\n\nA functional differential equation relates the derivative of an unknown function at time $t$ to its values at other times.\nOften these times fill a whole past interval.\nIt includes [[ordinary differential equations|0NT64JV96A]], delay equations and equations with distributed memory.\nThe state at time $t$ is the piece of the solution on an interval $[t - r, t]$, so the phase space is a space of functions.\nThe theory, developed by Krasovskii, Hale and others, extends existence, uniqueness and stability theory to this setting.\n\n## Explanations\n\n### L1 human\n\nIn an ordinary differential equation, the future depends only on the present state.\nIn a functional differential equation, it may depend on the recent past as well.\nFor instance, $x'(t) = -\\int_{t-1}^{t} x(s) \\, ds$ lets the rate of change depend on the average of the last time unit.\nTo start such an equation, one needs the whole history over that time unit.\n",
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