the solution on the interval is given by which is the solution to the inhomogeneous initial value problem with . This can be continued for the successive intervals by using the solution to the previous interval as inhomogeneous term. In practice, the initial value problem is often solved numerically. Example Suppose and . Then the initial value problem can be solved with integration, i.e., , where the initial condition is given by . Similarly, for the interval we integrate and fit the initial condition, i.e., Reduction to ODE In some cases, differential equations can be represented in a format that looks like delay differential equations. Example 1 Consider an equation Introduce to get a system of ODEs Example 2 An equation is equivalent to where The characteristic equation Similar to ODEs, many properties of linear DDEs can be characterized and analyzed using the characteristic equation. The characteristic equation associated with the linear DDE with discrete delays is The roots λ of the characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the spectrum. Because of the exponential in the characteristic equation, the DDE has, unlike the ODE case, an infinite number of eigenvalues, making a spectral analysis more involved. The spectrum does however have some properties which can be exploited in the analysis. For instance, even though there are an infinite number of eigenvalues, there are only a finite number of eigenvalues to the right of any vertical line in the complex plane. This characteristic equation is a nonlinear eigenproblem and there are many methods to compute the spectrum numerically. In some special situations it is possible to solve the characteristic equation explicitly. Consider, for example, the following DDE: The characteristic equation is There are an infinite number of solutions to this equation for complex λ. They are given by where Wk is the kth branch of the Lambert W function. Applications Dynamics of diabetes Epidemiology Population dynamics Classical electrodynamics See also Functional differential equation Halanay Inequality References Further reading External links Control theory Differential equations