Editing Differential equation (section)

===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.