= Delay differential equation
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A <delay differential equation> relates a derivative at the current argument to values at earlier arguments. For example, the <Buchstab function> obeys $\frac{d}{du}(u w(u))=w(u-1)$ 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.<Delay-differential equations by Skip Thompson>[Skip Thompson, Delay-differential equations, opening definition and equation (1): constant and state-dependent delays]{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.
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