interior of . If we are given a differential equation and the condition that when , then there is locally a solution to this problem if and are both continuous on . This solution exists on some interval with its center at . The solution may not be unique. (See Ordinary differential equation for other results.) However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order: such that For any nonzero , if and are continuous on some interval containing , is unique and exists. Related concepts A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times. An integro-differential equation (IDE) is an equation that combines aspects of a differential equation and an integral equation. A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion equations. A stochastic partial differential equation (SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications in quantum field theory and statistical mechanics. An ultrametric pseudo-differential equation is an equation which contains p-adic numbers in an ultrametric space. Mathematical models that involve ultrametric pseudo-differential equations use pseudo-differential operators instead of differential operators. A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form. Connection to difference equations The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation. Applications The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating