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→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 symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of 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.
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 symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of 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.
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。即使是非线性微分方程解的存在性、唯一性和可扩展性等基本问题，以及非线性偏微分方程初边值问题的适定性问题，也是一个难题（查阅，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。
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 [[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).