$$ \frac{dx}{dt} = \vec f(x(t), x(t-\tau), x(t-\tau_2), ..., x(t-\tau_n)) $$
Example:
First order DE
$$ \frac{dx}{ct} = -x \rightarrow x(t) = x_0e^{-t}$$
First order DDE:
$$\frac{dx}{dt} = -x(t-1)$$
This is a problem. We cannot use an initial value, we need **an initial history function (IHF)**. This is the behaviour of x(t) defined in an interval $[-\tau_0, 0]$, assuming solution time starts at $t=0$