An integro-differential equation is an equation involving both derivatives and integrals of an unknown function. A linear example of Volterra type is
 |
with an initial condition such as 
. The integral kernel 
,
the coefficient 
, and the forcing function 
are prescribed. The variable upper limit makes the derivative
at time 
depend on the earlier values of the solution, a mathematical model of memory.
In special cases, differentiation converts an integro-differential equation to an ordinary differential equation, but the original initial conditions must be retained. For example,
 |
together with 
, implies 
and 
, giving 
.
