Differential equations, ordinary, retarded

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An ordinary differential equation with deviating arguments (cf. Differential equations, ordinary, with distributed arguments) of delayed type, i.e. an equation in which the leading derivative of the sought function is defined, for any value of the argument, by the function itself and by lower derivatives taken for smaller or equal values of the argument. If time is taken as the argument, these equations and systems of these equations describe processes with an after-effect. The velocity of the state vector at any moment of time is determined by the state not only at that moment (as usual in the case of ordinary differential equations) but also at previous moments of time. Such a situation arises, in particular, in automatic control systems (cf. Automatic control theory) if there is a delay in the control mechanism.


Another often used phase for the equations is differential-delay equations.


[a1] J.K. Hale, "Functional differential equations" , Springer (1971)
[a2] R.D. Nusshaum, "Differential-delay equations with two time lags" , Amer. Math. Soc. (1978)
How to Cite This Entry:
Differential equations, ordinary, retarded. A.D. Myshkis (originator), Encyclopedia of Mathematics. URL:,_ordinary,_retarded&oldid=14979
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098