Editing Differential equation (section)

{{Short description|Type of functional equation (mathematics)}}
{{Distinguish|Difference equation}}
{{Differential equations}}

In [[mathematics]], a '''differential equation''' is an [[functional equation|equation]] that relates one or more unknown [[function (mathematics)|function]]s and their [[derivative]]s.<ref name="Zill2012">{{cite book|author=Dennis G. Zill|title=A First Course in Differential Equations with Modeling Applications|url=https://books.google.com/books?id=pasKAAAAQBAJ&q=%22ordinary+differential%22|date=15 March 2012|publisher=Cengage Learning|isbn=978-1-285-40110-2}}</ref> In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common in [[Mathematical model|mathematical models]] and [[Scientific law|scientific laws]]; therefore, differential equations play a prominent role in many disciplines including [[engineering]], [[physics]], [[economics]], and [[biology]].

The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.

Often when a [[closed-form expression]] for the solutions is not available, solutions may be approximated numerically using computers, and many [[numerical methods]] have been developed to determine solutions with a given degree of accuracy. The [[Dynamical systems theory|theory of dynamical systems]] analyzes the [[:wikt:qualitative|qualitative]] aspects of solutions, such as their [[Ergodic theory|average behavior]] over a long time interval.