In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function.

The general first-order, linear (only with respect to the term involving derivative) integro-differential equation is of the form

As is typical with differential equations, obtaining a closed-form solution can often be difficult. In the relatively few cases where a solution can be found, it is often by some kind of integral transform, where the problem is first transformed into an algebraic setting. In such situations, the solution of the problem may be derived by applying the inverse transform to the solution of this algebraic equation.

Consider the following second-order problem,

where

is the Heaviside step function. The Laplace transform is defined by,

Upon taking term-by-term Laplace transforms, and utilising the rules for derivatives and integrals, the integro-differential equation is converted into the following algebraic equation,

Thus,