Editing Differential equation (section)

==Types==
Differential equations can be classified several different ways. Besides describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

===Ordinary differential equations===
{{main|Ordinary differential equation|Linear differential equation}}
An [[ordinary differential equation]] (''ODE'') is an equation containing an unknown [[function of a real variable|function of one real or complex variable]] {{mvar|x}}, its derivatives, and some given functions of {{mvar|x}}. The unknown function is generally represented by a [[variable (mathematics)|variable]] (often denoted {{mvar|y}}), which, therefore, ''depends'' on {{mvar|x}}. Thus {{mvar|x}} is often called the [[independent variable]] of the equation. 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 equation]]s are the differential equations that are [[linear equation|linear]] in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of [[antiderivative|integrals]].

Most ODEs that are encountered in [[physics]] are linear. Therefore, most [[special functions]] may be defined as solutions of linear differential equations (see [[Holonomic function]]).

As, in general, the solutions of a differential equation cannot be expressed by a [[closed-form expression]], [[numerical ordinary differential equations|numerical methods]] are commonly used for solving differential equations on a computer.

===Partial differential equations===
{{main|Partial differential equation}}
A [[partial differential equation]] (''PDE'') is a differential equation that contains unknown [[Multivariable calculus|multivariable function]]s and their [[partial derivatives]]. (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 in closed form, or used to create a relevant [[computer model]].

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

===Non-linear differential equations===
{{main|Non-linear differential equations}}
A '''non-linear differential equation''' is a differential equation that is not a [[linear equation]] in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular [[Symmetry|symmetries]]. Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic of [[chaos theory|chaos]]. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. [[Navier–Stokes existence and smoothness]]). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.<ref>{{cite book
 | last1 = Boyce
 | first1 = William E.
 | last2 = DiPrima
 | first2 = Richard C.
 | title = Elementary Differential Equations and Boundary Value Problems
 | publisher =John Wiley & Sons
 | edition = 4th
 | year = 1967
 | pages = 3
 }}</ref>

Linear differential equations frequently appear as [[linearization|approximations]] to nonlinear equations. These approximations are only valid under restricted conditions. For example, the [[harmonic oscillator]] equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations.

===Equation order and degree{{anchor|Second order|Order}}===
The '''order of the differential equation''' is the highest ''[[order of derivative]]'' of the unknown function that appears in the differential equation. 
For example, an equation containing only [[first-order derivative]]s is a ''[[first-order differential equation]]'', an equation containing the [[second-order derivative]] is a ''second-order differential equation'', and so on.<ref>[[Eric W Weisstein|Weisstein, Eric W]]. "Ordinary Differential Equation Order." From [[MathWorld]]--A Wolfram Web Resource. http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html</ref><ref>{{usurped|1=[https://web.archive.org/web/20160401070512/http://www.kshitij-iitjee.com/Maths/Differential-Equations/order-and-degree-of-a-differential-equation.aspx Order and degree of a differential equation]}}, accessed Dec 2015.</ref>

When it is written as a [[polynomial equation]] in the unknown function and its derivatives, its '''degree of the differential equation''' is, depending on the context, the [[polynomial degree]] in the highest derivative of the unknown function,<ref>{{cite book |title=Elements of the Differential and Integral Calculus |author1=Elias Loomis |edition=revised |publisher=Harper & Bros. |year=1887 |isbn= |page=247 |url=https://books.google.com/books?id=DTI4AQAAMAAJ}} [https://books.google.com/books?id=DTI4AQAAMAAJ&pg=PA247 Extract of page 247]</ref> or its [[total degree]] in the unknown function and its derivatives. In particular, a [[linear differential equation]] has degree one for both meanings, but the non-linear differential equation <math>y'+y^2=0</math> is of degree one for the first meaning but not for the second one.

Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the [[thin-film equation]], which is a fourth order partial differential equation.

===Examples===

In the first group of examples ''u'' is an unknown function of ''x'', and ''c'' and ''ω'' are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between ''[[linear differential equation|linear]]'' and ''nonlinear'' differential equations, and between [[homogeneous differential equation|''homogeneous'' differential equation]]s and ''heterogeneous'' ones.
* Heterogeneous first-order linear constant coefficient ordinary differential equation:
*: <math> \frac{du}{dx} = cu+x^2. </math>
* Homogeneous second-order linear ordinary differential equation:
*: <math> \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. </math>
* Homogeneous second-order linear constant coefficient ordinary differential equation describing the [[harmonic oscillator]]:
*: <math> \frac{d^2u}{dx^2} + \omega^2u = 0. </math>
* Heterogeneous first-order nonlinear ordinary differential equation:
*: <math> \frac{du}{dx} = u^2 + 4. </math>
* Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a [[pendulum]] of length ''L'':
*: <math> L\frac{d^2u}{dx^2} + g\sin u = 0. </math>

In the next group of examples, the unknown function ''u'' depends on two variables ''x'' and ''t'' or ''x'' and ''y''.
* Homogeneous first-order linear partial differential equation:
*: <math> \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0. </math>
* Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the [[Laplace equation]]:
*: <math> \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. </math>
* Homogeneous third-order non-linear partial differential equation, the [[Korteweg–De Vries equation|KdV equation]]:
*: <math> \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. </math>