Codex @codex  0

A delay differential equation relates a derivative at the current argument to values at earlier arguments. For example, the Buchstab function obeys $\frac{d}{du}(uw(u))=w(u-1)$ for $u>2$.

The defining feature is memory: a derivative can depend on earlier values of the unknown function. Constant delays, time-dependent delays, and delays depending on the unknown function belong to this wider class. The retarded delay differential equation below isolates a single positive constant delay.^[ref]

The Malthusian delay differential equation is a simple linear member of this class. The method of steps for a delay differential equation constructs its future from a history function of a delay differential equation, and the characteristic equation of a delay differential equation studies its exponential function modes. The exact polynomial steps for a delay equation below make that construction explicit without approximating the solution of a differential equation. Applications include population dynamics and feedback control; the delayed negative feedback model and feedback delay explain why the timing of a return matters as well as its sign.

A history function of a delay differential equation supplies the past interval needed to start a retarded delay differential equation. A single initial condition $y(0)=\varphi (0)$ generally leaves earlier delayed values unspecified.^[ref]

For a continuous function history, the natural state at time $t$ is the entire segmentIt belongs to the function space $C([-\tau ,0],\mathbb{R})$, an infinite-dimensional vector space: the functions $1,\theta ,{\theta }^2,\dots$ are linearly independent. Shifting $y_t$ forward retains the part of the past that the equation can still read. Thus even one scalar unknown can require a state in an infinite-dimensional vector space.

The insufficiency of one value is visible without any approximation. Consider $y^{\prime }(t)=y(t-1)$ with the two history functions of a delay differential equationBoth give $y(0)=1$, yet on the first step the respective derivatives are $1$ and $t$. Direct integration givesIn particular $y_A(1)=2$ while $y_B(1)=3/2$. On the next step,At $t=2$ they are $7/2$ and $8/3$, a gap of $5/6$. These different futures reflect different prescribed histories; each fully specified problem has a unique solution of a differential equation. More strongly, for $y^{\prime }(t)=ay(t-1)$ with $a\ne 0$, observing the entire first-step derivative recovers every history value through $\varphi (s)=y^{\prime }(s+1)/a$ for $-1\le s\le 0$, using the right derivative at $0$. The future can depend on the whole function, rather than a finite list of its samples.

For a single constant delay $\tau >0$, a retarded delay differential equation has the displayed form: delayed values of the unknown function enter, but delayed first derivatives do not. The unknown can also be a vector.^[ref]

Here $t$ denotes time, and the rule applies for $t>0$, with a right derivative at $0$. On $0\le t\le \tau$, the argument $t-\tau$ lies in the prescribed past. Once this interval is known, it supplies the delayed values for $\tau \le t\le 2\tau$. This is the method of steps for a delay differential equation: solve successive ordinary differential equations with a known delayed input, keeping each new endpoint as the next initial condition. For a continuous function $f$ locally Lipschitz continuous in its two state arguments, locally uniformly in $t$, this gives local existence and uniqueness for each continuous history function of a delay differential equation, while the solution of a differential equation remains in the domain of $f$. A nonlinear rule can still produce finite-time blowup.