# Neutral differential equation

A differential equation with distributed argument (cf. Differential equations, ordinary, with distributed arguments) in which the highest derivative occurs for more than one value of the argument, among them a basic (untransformed) one, and this latter value is the largest of those present in the equation. For example, the equation

$$x'(t)=f(t,x(\alpha(t)),x'(\beta(t)))\label{*}\tag{*}$$

is a neutral differential equation when $\alpha(t)\leq t$, $\beta(t)\leq t$.

For a neutral differential equation the initial value problem is solvable; thus, if for \eqref{*} with increasing $\beta(t)$ one gives

$$x=\phi(t),\quad t\leq t_0,$$

then for

$$f\in C^{m,n,n},\quad\alpha,\beta\in C^m,\quad\phi\in C^p,\quad m,n\geq0,p\geq1,$$

there exists a (for $n\geq1$ unique) piecewise-smooth solution, which belongs to $C^k$ when $k=1+\min\{m,n,p-1\}$ compatibility conditions hold, that is, conditions of the type

$$\phi'(t_0)=f(t_0,\phi(\alpha(t_0)),\phi'(\beta(t_0))).$$

Neutral differential equations are one of the most thoroughly studied classes of equations with distributed arguments. They occur naturally in applied problems that contain in their statement some recurrence property.