A differential equation describes the relationship between unknown functions and their derivatives. The order of a differential equation describes the most times any function in it has been differentiated.

Given that *y* is a function of *x* and that *y´*, *y´´*, ...,
*y ^{(n)}* denote the derivatives

The problem of solving a differential equation is to find the function *y* whose derivatives satisfy the equation. For example, the differential equation *y´´* + *y* = 0 has the general solution *y* = *A* cos*x* + *B* sin*x*, where *A, B* are constants determined from boundary conditions. In the case where the
equations are linear, this can be done by breaking the original equation down into
smaller equations, solving those, and then adding the results back together. Unfortunately, many of the interesting differential equations are non-linear, which
means that they cannot be broken down in this way. There are also a number of techiques for solving differential equations using a computer including [finite-element methods]? and [relaxation techniques]?.

Ordinary differential equations are to be distinguished from **partial differential equations** where *y* is a function of several variables, and the differential equation involves [partial derivative]?s.

Differential equations are used to construct mathematical models of physical phenomena such as fluid dynamics or [celestial mechanics]?. Therefore, the study of differential equations is a wide field in both pure and applied mathematics.

Differential equations have intrinsically interesting properties such as whether or not solutions exist, and should solutions exist, whether those solutions are unique. Applied mathematicians, physicists and engineers are usually more interested in how to compute solutions to differential equations. These solutions are then used to design bridges, automobiles, aircraft, sewers, etc.