Integro-differential equation

Summary

An integro-differential equation involves both derivatives and integrals of the unknown function. A typical form is $x'(t) = f(t, x(t)) + \int_{t_0}^{t} K(t, s, x(s)) \, ds$, in which the integral describes memory of the past. Such equations appear in viscoelasticity, population dynamics with age structure, circuit theory and epidemiology. They are studied with the Laplace transform, by turning them into integral equations, or as functional differential equations.