In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with a functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs:

A general form of the time-delay differential equation for ${\displaystyle x(t)\in \mathbb {R} ^{n}}$ is $${\displaystyle {\frac {d}{dt}}x(t)=f(t,x(t),x_{t}),}$$ where ${\displaystyle x_{t}=\{x(\tau ):\tau \leq t\}}$ represents the trajectory of the solution in the past. In this equation, ${\displaystyle f}$ is a functional operator from ${\displaystyle \mathbb {R} \times \mathbb {R} ^{n}\times C^{1}(\mathbb {R} ,\mathbb {R} ^{n})}$ to ${\displaystyle \mathbb {R} ^{n}.}$

DDEs are mostly solved in a stepwise fashion with a principle called the method of steps. For instance, consider the DDE with a single delay $${\displaystyle {\frac {d}{dt}}x(t)=f(x(t),x(t-\tau ))}$$

with given initial condition ${\displaystyle \phi \colon [-\tau ,0]\to \mathbb {R} ^{n}}$. Then the solution on the interval ${\displaystyle [0,\tau ]}$ is given by ${\displaystyle \psi (t)}$ which is the solution to the inhomogeneous initial value problem $${\displaystyle {\frac {d}{dt}}\psi (t)=f(\psi (t),\phi (t-\tau )),}$$ with ${\displaystyle \psi (0)=\phi (0)}$. 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.

Suppose ${\displaystyle f(x(t),x(t-\tau ))=ax(t-\tau )}$ and ${\displaystyle \phi (t)=1}$. Then the initial value problem can be solved with integration,

$${\displaystyle x(t)=x(0)+\int _{s=0}^{t}{\frac {d}{dt}}x(s)\,ds=1+a\int _{s=0}^{t}\phi (s-\tau )\,ds,}$$

i.e., ${\displaystyle x(t)=at+1}$, where the initial condition is given by ${\displaystyle x(0)=\phi (0)=1}$. Similarly, for the interval ${\displaystyle t\in [\tau ,2\tau ]}$ we integrate and fit the initial condition,

$${\displaystyle {\begin{aligned}x(t)=x(\tau )+\int _{s=\tau }^{t}{\frac {d}{dt}}x(s)\,ds&=(a\tau +1)+a\int _{s=\tau }^{t}\left(a(s-\tau )+1\right)ds\\&=(a\tau +1)+a\int _{s=0}^{t-\tau }\left(as+1\right)ds,\end{aligned}}}$$