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  "body": "\n## Summary\n\nA neutral differential equation is a [[delay differential equation|39PCDRDB01]] in which the derivative of the unknown also appears with a delay.\nA typical form is $x'(t) = f(t, x(t), x(t - \\tau), x'(t - \\tau))$.\nSuch equations do not smooth their initial data over time, unlike retarded delay equations.\nThey arise in models of transmission lines, coupled oscillators and some population models.\n\n## Explanations\n\n### L1 human\n\nIn an ordinary delay equation, the rate of change depends on past states.\nIn a neutral equation, it also depends on past rates of change.\nFor $x'(t) = \\frac{1}{2} x'(t - 1) - x(t)$, a kink in the history is copied forward forever, only shrinking by the factor $\\frac{1}{2}$ each time unit.\nSo the solution never becomes smoother than its initial history.\n",
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