A delay differential equation (also called a differential delay equation or difference-differential equation, although the latter term has a different meaning in the modern literature) is a special type of functional differential equation. Delay differential equations are similar to ordinary differential equations, but their evolution involves past values of the state variable. The solution of delay differential equations therefore requires knowledge of not only the current state, but also of the state a certain time previously.
Examples include the equations defining the Dickman function
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(1)
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and the Buchstab function
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(2)
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(Panario 1998).
Hayes, A. "Delay-Differential Equations."