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.

See also

* Integrodifference equation

References

* [http://www.intmath.com/Laplace-transformation/9_Integro-differential-eqns-simultaneous-DE.php] at Interactive Mathematics
* Vangipuram Lakshmikantham, M. Rama Mohana Rao, ``Theory of Integro-Differential Equations",CRC Press, 1995