Editing Differential equation (section)

==Applications==

The study of differential equations is a wide field in [[pure mathematics|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 solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have [[closed-form expression|closed form]] solutions. Instead, solutions can be approximated using [[Numerical ordinary differential equations|numerical methods]].

Many fundamental laws of [[physics]] and [[chemistry]] can be formulated as differential equations. In [[biology]] and [[economics]], differential equations are used to [[mathematical modelling|model]] the behavior of [[complex system]]s. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order [[partial differential equation]], the [[wave equation]], which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by [[Joseph Fourier]], is governed by another second-order partial differential equation, the [[heat equation]]. It turns out that many [[diffusion]] processes, while seemingly different, are described by the same equation; the [[Black–Scholes]] equation in finance is, for instance, related to the heat equation.

The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See [[List of named differential equations]].