Definitions

# Integro-differential equation

An integro-differential equation is an equation which has both integrals and derivatives of an unknown function. The equation is of the form


frac{dx(t)}{dt} = f(t,x(t))+ int_{t_0}^t K(t,s,x(s)),ds where

x(t_0) = x_0, t_0 ge 0

For example:


frac{di}{dt} + 2i + 5int_{0}^{t}i,dt = u(t),quad i(0)=0.

An integro-differential equation is similar to a differential equation; therefore, tools such as the Laplace transform can be used to solve the equation.