Delay differential equation
The notion of delay differential equation (abbreviated DDE) is a variant of the notion of differential equation (in other words, delay differential equations are not (ordinary) differential equations).
First-order first-degree case
If we denote the dependent variable by and the independent variable by (Which we think of as time), the first-order first-degree case is:
The general case of a delay differential equation is of the form:
Note on autonomous case
The delay differential equations that we study are typically autonomous delay differential equations: an equation in the general form above is autonomous if, for any , the function is invariant under replacing by the function . Intuitively, what this means is that does not appear explicitly in , and all the behavior at previous points is specified in terms of how much earlier they were than .