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  "body": "\n## Definition\n\nLet $\\tau \u003e 0$ and $n \\ge 1$, and let $f \\colon \\R \\times \\R^n \\times \\R^n \\times \\R^n \\to \\R^n$ be a function.\nA *neutral differential equation* with delay $\\tau$ has the form\n$$x'(t) = f\\bigl(t, x(t), x(t - \\tau), x'(t - \\tau)\\bigr).$$\nHere $x$ is an unknown function with values in $\\R^n$, and $f$ depends on its last argument.\nAn initial value problem prescribes a history $\\phi$ on $[-\\tau, 0]$ that is continuously differentiable.\n\n## Remarks\n\nExample: $x'(t) = \\frac{1}{2} x'(t - 1) - x(t)$, with $f(t, u, v, w) = \\frac{1}{2} w - u$, which depends on $w$.\n\nNon-example: $x'(t) = -x(t - 1)$ is a [[delay differential equation|39PCDRDB01]] of retarded type, not a neutral one, since $f(t, u, v, w) = -v$ does not depend on $w$.\n\nA more general form, due to Hale, is $\\frac{d}{dt} D(t, x_t) = F(t, x_t)$, where $x_t$ is the segment of $x$ on $[t - \\tau, t]$ and $D(t, \\psi) = \\psi(0) - g(t, \\psi)$ for a suitable function $g$ of the past.\nThe linear case $\\frac{d}{dt}\\bigl(x(t) - c\\, x(t - \\tau)\\bigr) = a\\, x(t) + b\\, x(t - \\tau)$ is the standard test example.\n\n## Explanations\n\n### L1 human\n\nA neutral equation lets past derivatives, not only past values, drive the present derivative.\nThis feature makes its solutions keep the roughness of their initial data.\n",
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