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第214行： − + − − | last = Boyce − − | last = Boyce − − 最后的博伊斯 − − | first = William E. − − | first = William E. − − 首先是威廉姆 · e。 − − | last2 = DiPrima − − | last2 = DiPrima − − 2 DiPrima − − | first2 = Richard C. − − | first2 = Richard C. − − | first2 Richard c. − − | authorlink = − − | authorlink = − − 作者链接 − − | title = Elementary Differential Equations and Boundary Value Problems − − | title = Elementary Differential Equations and Boundary Value Problems − − 初等微分方程和边值问题 − − | publisher =John Wiley & Sons − − | publisher =John Wiley & Sons − − 出版商约翰威立 − − | series = − − | series = − − 系列 − − | volume = − − | volume = − − 音量 − − | edition = 4th − − | edition = 4th − − 第四版 − − | year = 1967 − − | year = 1967 − − 1967年 − − | location = − − | location = − − | 位置 − − | pages = 3 − − | pages = 3 − − 第三页 − − | language = − − | language = − − 语言 − − | url = − − | url = − − 我们会找到你的 − − | doi = − − | doi = − − 不会吧 − − | id = − − | id = − − 我会的 − − | isbn = − − | isbn = − − | isbn − − | mr = − − | mr = − − 先生 − − | zbl = − − | zbl = − − 我不会让你失望的 − − | jfm = }}</ref> − − | jfm = }}</ref> − − 参考文献 − − 第348行：
第220行： − 线性微分方程经常作为非线性方程的近似形式出现。这些近似只有在受限制的条件下才有效。例如，谐振子方程是非线性摆方程的近似，对于小幅度振荡是有效的(见下文)。+
→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.<ref>{{cite book
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.<ref>{{cite book
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(这里不考虑函数论元中的线性或非线性)。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。 文档{ cite book
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(这里不考虑函数论元中的线性或非线性)。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。
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).
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).
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).
线性微分方程经常作为非线性方程的近似形式出现。这些近似只有在受限制的条件下才有效。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===
==={{anchor|Second order}} Equation order===