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  "body": "\n## Definition\n\nLet $n, k \\ge 1$, let $0 \u003c \\tau_1 \u003c \\cdots \u003c \\tau_k$ be constants, and let $f \\colon \\R \\times (\\R^n)^{k+1} \\to \\R^n$ be a function.\nA *delay differential equation* with constant delays $\\tau_1, \\ldots, \\tau_k$ is an equation\n$$x'(t) = f\\bigl(t, x(t), x(t - \\tau_1), \\ldots, x(t - \\tau_k)\\bigr)$$\nfor an unknown function $x$ with values in $\\R^n$.\nLet $t_0 \\in \\R$ and let $\\phi \\colon [-\\tau_k, 0] \\to \\R^n$ be continuous, the *history*.\nA *solution* of the initial value problem is a continuous $x \\colon [t_0 - \\tau_k, t_0 + T) \\to \\R^n$, for some $T \u003e 0$, with these properties.\nIt equals the history: $x(t_0 + \\theta) = \\phi(\\theta)$ for $\\theta \\in [-\\tau_k, 0]$.\nIt is differentiable on $(t_0, t_0 + T)$, and the equation holds there.\n\n## Remarks\n\nExample: $x'(t) = -x(t - 1)$, with $n = k = 1$, $\\tau_1 = 1$ and $f(t, u, v) = -v$.\nIts characteristic equation, from $x = e^{\\lambda t}$, is $\\lambda = -e^{-\\lambda}$, which has infinitely many complex roots.\n\nNon-example: $x'(t) = x(t + 1)$ is not a delay equation, since the rate at time $t$ depends on a later time; such equations are called advanced.\nAn [[ordinary differential equation|0NT64JV96A]] $x' = f(t, x)$ is the degenerate case in which $f$ does not depend on the delayed arguments.\n\nDelay equations are a special case of [[functional differential equations|ZMCFEEPEKT]] of retarded type, with $F(t, x_t) = f(t, x_t(0), x_t(-\\tau_1), \\ldots, x_t(-\\tau_k))$.\nDelays may also vary with time or with the state, as in $x'(t) = f(t, x(t), x(t - \\tau(t)))$.\n\n## Explanations\n\n### L1 human\n\nThe derivative now depends on where the solution was some time ago.\nSo the start of the problem is a whole piece of past, not one point.\n",
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