Editing Equation (section)

==Differential equations==
{{main|Differential equation}}
[[File:Attracteur étrange de Lorenz.png|thumb|A [[strange attractor]], which arises when solving a certain [[differential equation]]]]
A [[differential equation]] is a [[mathematics|mathematical]] equation that relates some [[function (mathematics)|function]] with its [[derivative]]s. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in areas such as physics, chemistry, biology, and economics.

In [[pure mathematics]], differential equations are studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of [[dynamical systems]] puts emphasis on qualitative analysis of systems described by differential equations, while many [[numerical methods]] have been developed to determine solutions with a given degree of accuracy.

===Ordinary differential equations===
{{main|Ordinary differential equation}}
An [[ordinary differential equation]] or ODE is an equation containing a function of one [[independent variable]] and its derivatives. The term "''ordinary''" is used in contrast with the term [[partial differential equation]], which may be with respect to ''more than'' one independent variable.

Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by [[elementary functions]] in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and [[numerical ordinary differential equations|numerical]] methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.

===Partial differential equations===
{{main|Partial differential equation}}
A [[partial differential equation]] (PDE) is a [[differential equation]] that contains unknown [[Multivariable calculus|multivariable functions]] and their [[partial derivative]]s. (This is in contrast to [[ordinary differential equations]], which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant [[computer model]].

PDEs can be used to describe a wide variety of phenomena such as [[sound]], [[heat]], [[electrostatics]], [[electrodynamics]], [[fluid flow]], [[Elasticity (physics)|elasticity]], or [[quantum mechanics]]. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional [[dynamical systems]], partial differential equations often model [[multidimensional systems]]. PDEs find their generalisation in [[stochastic partial differential equations]].