Ordinary differential equations Definition

Neutral differential equation

Definition

Let $\tau > 0$ and $n \ge 1$ , and let $f \colon \R \times \R^n \times \R^n \times \R^n \to \R^n$ be a function. A neutral differential equation with delay $\tau$ has the form $x'(t) = f\bigl(t, x(t), x(t - \tau), x'(t - \tau)\bigr).$ Here $x$ is an unknown function with values in $\R^n$ , and $f$ depends on its last argument. An initial value problem prescribes a history $\phi$ on $[-\tau, 0]$ that is continuously differentiable.

Remarks

Example: $x'(t) = \frac{1}{2} x'(t - 1) - x(t)$ , with $f(t, u, v, w) = \frac{1}{2} w - u$ , which depends on $w$ .

Non-example: $x'(t) = -x(t - 1)$ is a delay differential equation of retarded type, not a neutral one, since $f(t, u, v, w) = -v$ does not depend on $w$ .

A 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. The 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.

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Delay differential equation concept
Neutral differential equation definition

Tag	`7E9H1SPAK8`
URI	`m2://7E9H1SPAK8`
Kind	definition
Writing	W0 stub
MSC	34K40, 34K05
Defines	Neutral differential equation
Uses	Delay differential equation
Changed	2 November 2026

Neutral differential equation

Definition

Remarks

Reading path

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