In mathematics, an integrodifferential equation is an equation that involves both integrals and derivatives of a function.
General first order linear equations
The general firstorder, linear integrodifferential equation is of the form

\frac{d}{dx}u(x) + \int_{x_0}^x f(t,u(t))\,dt = g(x,u(x)), \qquad u(x_0) = u_0, \qquad x_0 \ge 0.
As is typical with differential equations, obtaining a closedform 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.
Example
Consider the following firstorder problem,

u'(x) + 2u(x) + 5\int_{0}^{x}u(t)\,dt = \left\{ \begin{array}{ll} 1, \qquad x \geq 0\\ 0, \qquad x < 0 \end{array} \right. \qquad \text{with} \qquad u(0)=0.
The Laplace transform is defined by,

U(s) = \mathcal{L} \left\{u(x)\right\}=\int_0^{\infty} e^{sx} u(x) \,dx.
Upon taking termbyterm Laplace transforms, and utilising the rules for derivatives and integrals, the integrodifferential equation is converted into the following algebraic equation,

s U(s)  u(0) + 2U(s) + \frac{5}{s}U(s) = \frac{1}{s}.
Thus,

U(s) = \frac{1}{s^2 + 2s + 5} .
Inverting the Laplace transform using contour integral methods then gives

u(x) = \frac{1}{2} e^{x} \sin(2x) .
Applications
Integrodifferential equations model many situations from science and engineering. A particularly rich source is electrical circuit analysis.
The activity of interacting inhibitory and excitatory neurons can be described by a system of integrodifferential equations, see for example the WilsonCowan model.
See also
External links

Interactive Mathematics

Numerical solution of the example using Chebfun
References

Vangipuram Lakshmikantham, M. Rama Mohana Rao, “Theory of IntegroDifferential Equations”, CRC Press, 1995
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