= Retarded delay differential equation
{title2=$y'(t)=f(t,y(t),y(t-\tau))$}
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>.<Delay-differential equations by Skip Thompson>[Skip Thompson, Delay-differential equations, opening definition and equation (1): delayed values versus delayed derivatives]{ref}
Here $t$ denotes time, and the rule applies for $t>0$, with a right <derivative> at $0$. On $0\leq t\leq\tau$, the argument $t-\tau$ lies in the prescribed past. Once this interval is known, it supplies the delayed values for $\tau\leq t\leq2\tau$. 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>.
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