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第88行： − 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 behavior 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. − − 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 (see below). − − Linear differential equations frequently appear as 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 (see below). − + − 方程的阶 − − Differential equations are described by their order, determined by the term with the [[Derivative#Higher derivatives|highest derivatives]]. An equation containing only first derivatives is a ''first-order differential equation'', an equation containing the [[second 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>[http://www.kshitij-iitjee.com/Maths/Differential-Equations/order-and-degree-of-a-differential-equation.aspx Order and degree of a differential equation] {{Webarchive|url=https://web.archive.org/web/20160401070512/http://www.kshitij-iitjee.com/Maths/Differential-Equations/order-and-degree-of-a-differential-equation.aspx |date=2016-04-01 }}, accessed Dec 2015.</ref> 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|thin film equation]], which is a fourth order partial differential equation. − + − 微分方程的阶数是由它们的导数的最高阶决定的。只含有一阶导数的方程是一阶微分方程，含有二阶导数的方程是二阶微分方程，等等。描述自然现象的微分方程几乎总是只有一阶和二阶导数，但也有一些例外，例如薄膜方程，它是一个四阶偏微分方程。+ − ===Examples=== − 示例
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===非线性微分方程===
===非线性微分方程===
非线性微分方程是微分方程的一种，但它不是关于未知函数及其导数的线性方程(这里不考虑函数本身的线性或非线性)。能够精确求解非线性微分方程的方法很少; 那些已有的方法通常依赖于方程具有某种特定的对称性。非线性微分方程在更长的时间段内表现出非常复杂的行为，具有混沌特性。即使非线性微分方程也有解的存在性、唯一性和可扩展性等基本问题以及初边值问题的适定性问题，但对其研究也是一个难题（可参考纳维-斯托克斯方程的存在性和光滑性)。然而，如果微分方程是一个有意义物理过程的正确表述，那么人们期望它有一个解析解。
非线性微分方程是微分方程的一种，但它不是关于未知函数及其导数的线性方程(这里不考虑函数本身的线性或非线性)。能够精确求解非线性微分方程的方法很少; 那些已有的方法通常依赖于方程具有某种特定的对称性。非线性微分方程在更长的时间段内表现出非常复杂的行为，具有混沌特性。即使非线性微分方程也有解的存在性、唯一性和可扩展性等基本问题以及初边值问题的适定性问题，但对其研究也是一个难题（可参考纳维-斯托克斯方程的存在性和光滑性)。然而，如果微分方程是一个有意义物理过程的正确表述，那么人们期望它有一个解析解。
线性微分方程经常作为非线性方程的近似形式出现。这些近似只有某些限制条件下才有效。例如，谐振子方程是非线性摆方程的近似这一情况只有对于小幅度振荡是有效的(见下文)。
线性微分方程经常作为非线性方程的近似形式出现。这些近似只有某些限制条件下才有效。例如，谐振子方程是非线性摆方程的近似这一情况只有对于小幅度振荡是有效的(见下文)。
==={{anchor|Second order}} Equation order===
===方程的阶===
Differential equations are described by their order, determined by the term with the highest derivatives. An equation containing only first derivatives is a first-order differential equation, an equation containing the second derivative is a second-order differential equation, and so on. 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.
微分方程的阶数是由它们的导数的最高阶决定的。只含有一阶导数的方程是一阶微分方程，含有二阶导数的方程是二阶微分方程，等等。描述自然现象的微分方程几乎总是只有一阶和二阶导数<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>[http://www.kshitij-iitjee.com/Maths/Differential-Equations/order-and-degree-of-a-differential-equation.aspx Order and degree of a differential equation] {{Webarchive|url=https://web.archive.org/web/20160401070512/http://www.kshitij-iitjee.com/Maths/Differential-Equations/order-and-degree-of-a-differential-equation.aspx |date=2016-04-01 }}, accessed Dec 2015.</ref>，但也有一些例外，例如薄膜方程，它是一个四阶偏微分方程。
===示例===