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  "body": "\n## Definition\n\nLet $r \\ge 0$ and $n \\ge 1$, and let $C = C([-r, 0], \\R^n)$ be the space of continuous functions with the supremum norm.\nFor a continuous function $x$ defined on an interval containing $[t - r, t]$, let $x_t \\in C$ be the segment $x_t(\\theta) = x(t + \\theta)$ for $\\theta \\in [-r, 0]$.\nLet $\\Omega \\subseteq \\R \\times C$ be open and let $F \\colon \\Omega \\to \\R^n$.\nA *(retarded) functional differential equation* is the equation\n$$x'(t) = F(t, x_t).$$\nA *solution* through $(\\sigma, \\phi) \\in \\Omega$ is a continuous $x \\colon [\\sigma - r, \\sigma + T) \\to \\R^n$, for some $T \u003e 0$, such that $x_\\sigma = \\phi$, $(t, x_t) \\in \\Omega$ and $x'(t) = F(t, x_t)$ for $t \\in (\\sigma, \\sigma + T)$.\n\n## Remarks\n\nExample: $x'(t) = -\\int_{-1}^{0} x(t + \\theta) \\, d\\theta$ is of this form with $r = 1$ and $F(t, \\psi) = -\\int_{-1}^{0} \\psi(\\theta) \\, d\\theta$.\n\nNon-example: $x'(t) = x(t + 1)$ is not of this form, since $x'(t)$ depends on the value at the later time $t + 1$, which is not part of $x_t$.\n\nFor $r = 0$, $C$ is $\\R^n$ and the equation is an [[ordinary differential equation|0NT64JV96A]].\nIf $F$ is continuous and locally Lipschitz in its second argument, the [[initial value problem|0FXDW992DS]] has a unique local solution through each $(\\sigma, \\phi) \\in \\Omega$.\nEquations whose derivative term also involves the past, such as $\\frac{d}{dt}\\bigl(x(t) - c\\, x(t - r)\\bigr) = F(t, x_t)$, are called neutral.\n\n## Explanations\n\n### L1 human\n\nThe derivative at time $t$ is any rule that reads the solution on the window $[t - r, t]$.\nThe window slides forward in time and plays the role of the state.\n",
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