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.
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, Inverting the Laplace transform using contour integral methods then gives Alternatively, one can complete the square and use a table of Laplace transforms ("exponentially decaying sine wave") or recall from memory to proceed: Integro-differential equations model many situations from science and engineering, such as in circuit analysis.
[1] The activity of interacting inhibitory and excitatory neurons can be described by a system of integro-differential equations, see for example the Wilson-Cowan model.
The Whitham equation is used to model nonlinear dispersive waves in fluid dynamics.
[2] Integro-differential equations have found applications in epidemiology, the mathematical modeling of epidemics, particularly when the models contain age-structure[3] or describe spatial epidemics.
[4] The Kermack-McKendrick theory of infectious disease transmission is one particular example where age-structure in the population is incorporated into the modeling framework.