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 dy/dx, d2y/dx2, ..., dny/dxn, an ordinary differential equation is an equation involving x, y, y´, y´´, .... The order of a differential equation is the the order n of the highest derivative that appears. An important special case is when the equations do not involve x. These kind of differential equations have the property that the space can be divided into equivalence classes based on whether two points lie on the same solution. These differential equations are vector fields. Since the laws of physics are not believed to change with time, the world is governed by such differential equations. See also Symplectic topology.
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 cosx + B sinx, 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 derivatives.
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.