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第2行： − + + + + + + − {{Distinguish|Difference equation}} − − [[File:Elmer-pump-heatequation.png|thumb|350px|Visualization of heat transfer in a pump casing, created by solving the [[heat equation]]. [[Heat]] is being generated internally in the casing and being cooled at the boundary, providing a [[steady state]] temperature distribution.]] − − Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution. − − 通过求解热力学方程，我们建立了泵壳内传热的可视化模型。热量在内部产生并在边界冷却，从而为整体提供稳定的温度分布。 − − In mathematics, a '''differential equation''' is an [[equation]] that relates one or more [[function (mathematics)|function]]s and their [[derivative]]s.<ref name="Zill2012">{{cite book|author=Dennis G. Zill|title=A First Course in Differential Equations with Modeling Applications|url=https://books.google.com/books?id=pasKAAAAQBAJ&printsec=frontcover#v=snippet&q=%22ordinary%20differential%22&f=false|date=15 March 2012|publisher=Cengage Learning|isbn=1-285-40110-7}}</ref> In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common, therefore differential equations play a prominent role in many disciplines including [[engineering]], [[physics]], [[economics]], and [[biology]]. − − In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common, therefore differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. − − 在数学上，'''<font color="#ff8000">微分方程 Differential Equation</font><font>'''是可以将一个或多个函数及其导数相互关联的方程。在实际应用中，函数通常代表物理量，导数代表其变化率，而微分方程则定义了两者之间的关系。由于这种关系十分普遍，因此微分方程在包括工程学、物理学、经济学和生物学在内的许多学科中得到了广泛的应用。 − − − − Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. − − Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. − − − Often when a [[closed-form expression]] for the solutions is not available, solutions may be approximated numerically using computers. The theory of [[dynamical systems]] puts emphasis on [[:wikt:qualitative|qualitative]] analysis of systems described by differential equations, while many [[numerical methods]] have been developed to determine solutions with a given degree of accuracy. − − Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. 第36行：
第17行： − + − 历史+ − − − Differential equations first came into existence with the [[History of calculus|invention of calculus]] by [[Isaac Newton|Newton]] and [[Leibniz]]. In Chapter 2 of his 1671 work [[Method of Fluxions|''Methodus fluxionum et Serierum Infinitarum'']],<ref>Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].</ref> Isaac Newton listed three kinds of differential equations: − − Differential equations first came into existence with the invention of calculus by Newton and Leibniz. In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum, Isaac Newton listed three kinds of differential equations: − − 微分方程是在牛顿和莱布尼茨发明微积分后才出现的。艾萨克 · 牛顿在他1671年的著作《无限的循环与系列》的第二章中列出了三种微分方程: 第56行：
第30行： − In all these cases, {{mvar|y}} is an unknown function of {{mvar|x}} (or of <math>x_1</math> and <math>x_2</math>), and {{mvar|f}} is a given function. − − In all these cases, {{mvar|y}} y is an unknown function of (or of <math>x_1</math> and <math>x_2</math>), and is a given function. − − − He solves these examples and others using infinite series and discusses the non-uniqueness of solutions. − − He solves these examples and others using infinite series and discusses the non-uniqueness of solutions. − + − [[Jacob Bernoulli]] proposed the [[Bernoulli differential equation]] in 1695.<ref>{{Citation | last1=Bernoulli | first1=Jacob | author1-link=Jacob Bernoulli | title=Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis | year=1695 | journal=[[Acta Eruditorum]]}}</ref> This is an [[ordinary differential equation]] of the form − − Jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is an ordinary differential equation of the form − − 雅可比·伯努利在1695年提出了伯努利微分方程。这种方程是'''<font color="#ff8000">常微分方程 Ordinary Differential Equation</font><font>'''的一种形式， − 第83行：
第43行： − for which the following year Leibniz obtained solutions by simplifying it.<ref>{{Citation | last1=Hairer | first1=Ernst | last2=Nørsett | first2=Syvert Paul | last3=Wanner | first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-56670-0 | year=1993}}</ref>+ − − for which the following year Leibniz obtained solutions by simplifying it. − − 莱布尼茨于第二年将方程简化从而得到了方程的解。 − − − − Historically, the problem of a vibrating string such as that of a [[musical instrument]] was studied by [[Jean le Rond d'Alembert]], [[Leonhard Euler]], [[Daniel Bernoulli]], and [[Joseph-Louis Lagrange]].<ref>{{cite journal|url = http://homes.chass.utoronto.ca/~cfraser/vibration.pdf |title = Review of ''The evolution of dynamics, vibration theory from 1687 to 1742'', by John T. Cannon and Sigalia Dostrovsky|last= Frasier|first=Craig|journal=Bulletin (New Series) of the American Mathematical Society |date=July 1983 |volume= 9| issue = 1}}</ref><ref>{{cite journal |first=Gerard F. |last=Wheeler |first2=William P. |last2=Crummett |title=The Vibrating String Controversy |journal= [[American Journal of Physics|Am. J. Phys.]] |year=1987 |volume=55 |issue=1 |pages=33–37 |doi=10.1119/1.15311 |bibcode = 1987AmJPh..55...33W }}</ref><ref>For a special collection of the 9 groundbreaking papers by the three authors, see [http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage= First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings] (retrieved 13 Nov 2012). Herman HJ Lynge and Son.</ref><ref>For de Lagrange's contributions to the acoustic wave equation, can consult [https://books.google.com/books?id=D8GqhULfKfAC&pg=PA18 Acoustics: An Introduction to Its Physical Principles and Applications] Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)</ref> In 1746, d’Alembert discovered the one-dimensional [[wave equation]], and within ten years Euler discovered the three-dimensional wave equation.<ref name=Speiser>Speiser, David. ''[https://books.google.com/books?id=9uf97reZZCUC&pg=PA191 Discovering the Principles of Mechanics 1600-1800]'', p. 191 (Basel: Birkhäuser, 2008).</ref> − − Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation. − − 历史上，让·勒朗·达朗贝尔，欧拉，丹尼尔·伯努利和约瑟夫·路易斯·拉格朗日等都研究过弦（比如乐器的弦）振动问题。1746年，达朗贝尔发现了一维波动方程，十年之内，欧拉又发现了三维波动方程。 − − − − The [[Euler–Lagrange equation]] was developed in the 1750s by Euler and Lagrange in connection with their studies of the [[tautochrone]] problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to [[mechanics]], which led to the formulation of [[Lagrangian mechanics]]. − The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. − 欧拉-拉格朗日方程式是欧拉和拉格朗日在18世纪50年代结合他们对等时降线问题的研究而发明的。这是一个不考虑起始点的曲线求解问题，其中一个加权的粒子将在一个给定的时间内下降到一个固定的点。拉格朗日在1755年解决了这个问题，并将其寄给欧拉。二人都进一步发展了拉格朗日的方法并将其应用于力学，从而促使了拉格朗日力学的形成。+ − − In 1822, [[Joseph Fourier|Fourier]] published his work on [[heat flow]] in ''Théorie analytique de la chaleur'' (The Analytic Theory of Heat),<ref>{{Cite book | last = Fourier | first = Joseph | title = Théorie analytique de la chaleur | publisher = Firmin Didot Père et Fils | year = 1822 | location = Paris | language = French | url=https://archive.org/details/bub_gb_TDQJAAAAIAAJ | oclc=2688081 }}</ref> in which he based his reasoning on [[Newton's law of cooling]], namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his [[heat equation]] for conductive diffusion of heat. This partial differential equation is now taught to every student of mathematical physics.+ − In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat), in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his heat equation for conductive diffusion of heat. This partial differential equation is now taught to every student of mathematical physics. − 1822年，傅立叶在《热的分析理论》中发表了他关于热流的研究成果，其中他以牛顿的冷却定律为基础进行推导，即两个相邻分子之间的热流与它们之间微小的温差成正比。这本书中包含了傅立叶关于热传导扩散的热方程式的建议。现在，每一个学习数学物理的学生都需要学习这类偏微分方程。+ − ==Example== − 示例 − − − In [[classical mechanics]], the motion of a body is described by its position and velocity as the time value varies. [[Newton's laws of motion|Newton's laws]] allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time. − − In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time. + − − − In some cases, this differential equation (called an [[equations of motion|equation of motion]]) may be solved explicitly. − − In some cases, this differential equation (called an equation of motion) may be solved explicitly. − − − An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity. − − An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity. − ==Types== − 微分方程的类型 − − − Differential equations can be divided into several types. Apart from 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/Partial, Linear/Non-linear, and Homogeneous/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. − − Differential equations can be divided into several types. Apart from 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/Partial, Linear/Non-linear, and Homogeneous/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. + − + − + − {{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. − − An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable , its derivatives, and some given functions of . The unknown function is generally represented by a 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. − − 常微分方程是只含有一个实变量或复变量的未知函数，其导数以及此函数的一些方程。未知函数因变量(通常由 {{mvar|y}} 表示)，其常常随 {{mvar|x}}的变化而变化 。因此 {{mvar|x}} 通常被称为方程式的自变量。“常微分方程”一词与偏微分方程一词相比，后者涉及一个以上的独立变量。 − − − [[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]]. − − Linear differential equations are the differential equations that are 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 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]]). − − 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. − − As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical methods are commonly used for solving differential equations on a computer. − ===Partial differential equations=== − 偏微分方程 − {{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]]. − − A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions 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. − − '''<font color="#ff8000">偏微分方程 Partial Differential Equation</font><font>'''是一种包含多元函数及其偏导数的微分方程函数(这与处理单变量函数及其导数的常微分方程不同)。偏微分方程可用于描述涉及多元函数的问题求闭式解，或者用于创建相关的计算机模型。 − − − − 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]]. − − PDEs can be used to describe a wide variety of phenomena in nature such as sound, heat, electrostatics, electrodynamics, fluid flow, 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=== − 非线性微分方程 − − − {{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 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.
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{{short description|Mathematical equation involving derivatives of an unknown function}}
{{#seo:
|keywords=数学，方程
|description=将一个或多个函数及其导数相互关联的方程。
}}
[[File:Elmer-pump-heatequation.png|thumb|350px|通过求解热力学方程，我们建立了泵壳内传热的可视化模型。热量在内部产生并在边界冷却，从而为整体提供稳定的温度分布。]]
在数学上，'''微分方程 Differential Equation'''是可以将一个或多个函数及其导数相互关联的方程。<ref name="Zill2012">{{cite book|author=Dennis G. Zill|title=A First Course in Differential Equations with Modeling Applications|url=https://books.google.com/books?id=pasKAAAAQBAJ&printsec=frontcover#v=snippet&q=%22ordinary%20differential%22&f=false|date=15 March 2012|publisher=Cengage Learning|isbn=1-285-40110-7}}</ref>在实际应用中，函数通常代表物理量，导数代表其变化率，而微分方程则定义了两者之间的关系。由于这种关系十分普遍，因此微分方程在包括工程学、物理学、经济学和生物学在内的许多学科中得到了广泛的应用。
微分方程的研究主要包括对微分方程解(满足每个方程的函数集)及其解的性质的研究。只有最简单的微分方程才能直接用公式求解；然而，有时无需精确计算便可以确定给定微分方程的解的许多性质。
微分方程的研究主要包括对微分方程解(满足每个方程的函数集)及其解的性质的研究。只有最简单的微分方程才能直接用公式求解；然而，有时无需精确计算便可以确定给定微分方程的解的许多性质。
一般地，当闭式解不存在时，可以用计算机求方程的近似解。动力系统理论着重于对由微分方程描述的系统进行定性分析。同时，现在已经得出了许多数值方法来计算给定精度下微分方程的解。
一般地，当闭式解不存在时，可以用计算机求方程的近似解。动力系统理论着重于对由微分方程描述的系统进行定性分析。同时，现在已经得出了许多数值方法来计算给定精度下微分方程的解。
==History==
==历史==
微分方程是在牛顿和莱布尼茨发明微积分后才出现的。Isaac Newton 艾萨克·牛顿在他1671年的著作《无限的循环与系列 Method of Fluxions》的第二章<ref>Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].</ref>中列出了三种微分方程:
</math>
</math>
在这些例子中，{{mvar|y}}是自变量 {{mvar|x}}(或者是<math>x_1</math> 和 <math>x_2</math>)的未知函数，并且 {{mvar|f}} 是一个给定的函数。
在这些例子中，{{mvar|y}}是自变量 {{mvar|x}}(或者是<math>x_1</math> 和 <math>x_2</math>)的未知函数，并且 {{mvar|f}} 是一个给定的函数。
他利用无穷级数来求解这些以及其他例子，并讨论了解的非唯一性。
他利用无穷级数来求解这些以及其他例子，并讨论了解的非唯一性。
雅可比·伯努利 Jacob Bernoulli在1695年提出了伯努利微分方程。<ref>{{Citation | last1=Bernoulli | first1=Jacob | author1-link=Jacob Bernoulli | title=Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis | year=1695 | journal=[[Acta Eruditorum]]}}</ref>这种方程是'''常微分方程 Ordinary Differential Equation'''的一种形式，
莱布尼茨 Leibniz于第二年将方程简化从而得到了方程的解。<ref>{{Citation | last1=Hairer | first1=Ernst | last2=Nørsett | first2=Syvert Paul | last3=Wanner | first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-56670-0 | year=1993}}</ref>
历史上，让·勒朗·达朗贝尔 Jean le Rond d'Alembert，欧拉 Leonhard Euler，丹尼尔·伯努利 Daniel Bernoulli和约瑟夫·路易斯·拉格朗日 Joseph-Louis Lagrange等都研究过弦（比如乐器的弦）振动问题。<ref>{{cite journal|url = http://homes.chass.utoronto.ca/~cfraser/vibration.pdf |title = Review of ''The evolution of dynamics, vibration theory from 1687 to 1742'', by John T. Cannon and Sigalia Dostrovsky|last= Frasier|first=Craig|journal=Bulletin (New Series) of the American Mathematical Society |date=July 1983 |volume= 9| issue = 1}}</ref><ref>{{cite journal |first=Gerard F. |last=Wheeler |first2=William P. |last2=Crummett |title=The Vibrating String Controversy |journal= [[American Journal of Physics|Am. J. Phys.]] |year=1987 |volume=55 |issue=1 |pages=33–37 |doi=10.1119/1.15311 |bibcode = 1987AmJPh..55...33W }}</ref><ref>For a special collection of the 9 groundbreaking papers by the three authors, see [http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage= First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings] (retrieved 13 Nov 2012). Herman HJ Lynge and Son.</ref><ref>For de Lagrange's contributions to the acoustic wave equation, can consult [https://books.google.com/books?id=D8GqhULfKfAC&pg=PA18 Acoustics: An Introduction to Its Physical Principles and Applications] Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)</ref> 1746年，达朗贝尔发现了一维波动方程，十年之内，欧拉又发现了三维波动方程。<ref name=Speiser>Speiser, David. ''[https://books.google.com/books?id=9uf97reZZCUC&pg=PA191 Discovering the Principles of Mechanics 1600-1800]'', p. 191 (Basel: Birkhäuser, 2008).</ref>
==[[用户:Yuling|Yuling]]（[[用户讨论:Yuling|讨论]]） independent of the starting point 这里翻译不太好。
欧拉-拉格朗日方程式 Euler–Lagrange equation是欧拉和拉格朗日在18世纪50年代结合他们对等时降线问题的研究而发明的。这是一个不考虑起始点的曲线求解问题，其中一个加权的粒子将在一个给定的时间内下降到一个固定的点。拉格朗日在1755年解决了这个问题，并将其寄给欧拉。二人都进一步发展了拉格朗日的方法并将其应用于力学，从而促使了拉格朗日力学的形成。
1822年，Joseph Fourier 傅立叶在《热的分析理论 Théorie analytique de la chaleur》中发表了他关于热流的研究成果，<ref>{{Cite book | last = Fourier | first = Joseph | title = Théorie analytique de la chaleur | publisher = Firmin Didot Père et Fils | year = 1822 | location = Paris | language = French | url=https://archive.org/details/bub_gb_TDQJAAAAIAAJ | oclc=2688081 }}</ref>其中他以[[牛顿的冷却定律 Newton's law of cooling]]为基础进行推导，即两个相邻分子之间的热流与它们之间微小的温差成正比。这本书中包含了傅立叶关于热传导扩散的热方程式的建议。现在，每一个学习数学物理的学生都需要学习这类偏微分方程。
==示例==
在经典力学中，物体运动是由其不断随时间变化的位置和速度来描述的。这些变量的表达在牛顿定律中是动态的(给定位置、速度、加速度和作用在物体上的各种力) ，并以时间函数的形式给出了未知物体位置的微分方程。
在经典力学中，物体运动是由其不断随时间变化的位置和速度来描述的。这些变量的表达在牛顿定律中是动态的(给定位置、速度、加速度和作用在物体上的各种力) ，并以时间函数的形式给出了未知物体位置的微分方程。
在某些情况下，这种微分方程(称为运动方程)可以精确地求解。
在某些情况下，这种微分方程(称为运动方程)可以精确地求解。
使用微分方程来模拟现实世界问题的一个例子是仅考虑重力和空气阻力来确定球在空中落下的速度。球对地面的加速度是重力加速度减去由于空气阻力提供的加速度。重力被认为是常数，空气阻力可以被模拟为与球的速度成正比。这意味着球的加速度，也就是其速度的导数，取决于速度(而速度取决于时间)。找到时间的函数--速度--需要解决一个微分方程问题并验证其正确性。
使用微分方程来模拟现实世界问题的一个例子是仅考虑重力和空气阻力来确定球在空中落下的速度。球对地面的加速度是重力加速度减去由于空气阻力提供的加速度。重力被认为是常数，空气阻力可以被模拟为与球的速度成正比。这意味着球的加速度，也就是其速度的导数，取决于速度(而速度取决于时间)。找到时间的函数--速度--需要解决一个微分方程问题并验证其正确性。
==微分方程的类型==
微分方程可分为以下几种类型。除了描述方程本身的性质之外，微分方程的多种类型为我们选择何种解决方案提供了多种指导。常见的微分方程有: 常微分/偏微分方程、线性/非线性方程和齐次/非齐次方程。微分方程还有许多类型，以及许多在特定的情况下实用的其它性质和子类。
微分方程可分为以下几种类型。除了描述方程本身的性质之外，微分方程的多种类型为我们选择何种解决方案提供了多种指导。常见的微分方程有: 常微分/偏微分方程、线性/非线性方程和齐次/非齐次方程。微分方程还有许多类型，以及许多在特定的情况下实用的其它性质和子类。
===Ordinary differential equations===
===常微分方程===
常微分方程
'''常微分方程 ordinary differential equation(ODE)'''是只含有一个实变量或复变量的未知函数，其导数以及此函数的一些方程。未知函数因变量(通常由 {{mvar|y}} 表示)，其常常随 {{mvar|x}}的变化而变化 。因此 {{mvar|x}} 通常被称为方程式的自变量。“常微分方程”一词与偏微分方程一词相比，后者涉及一个以上的独立变量。
线性微分方程是指方程中未知函数及其导数都是线性的微分方程。关于这些方程的理论发展得很好，在多数情况下可以用积分来表示它们的解。
线性微分方程是指方程中未知函数及其导数都是线性的微分方程。关于这些方程的理论发展得很好，在多数情况下可以用积分来表示它们的解。
物理学中遇到的大多数常微分方程都是线性的。因此，大多数特殊函数可以定义为线性微分方程的解(见完整性函数)。
物理学中遇到的大多数常微分方程都是线性的。因此，大多数特殊函数可以定义为线性微分方程的解(见完整性函数)。
一般地，微分方程的解不能用解析解表示，而会在计算机上利用数值方法求解。
一般地，微分方程的解不能用解析解表示，而会在计算机上利用数值方法求解。
===偏微分方程===
'''偏微分方程 Partial Differential Equation(PDE)'''是一种包含多元函数及其偏导数的微分方程函数(这与处理单变量函数及其导数的常微分方程不同)。偏微分方程可用于描述涉及多元函数的问题求闭式解，或者用于创建相关的计算机模型。
偏微分方程可以用来描述自然界中各种各样的现象，如声音、热量、静电、电动力学、流体流动、弹性和量子力学等。这些看起来截然不同的物理现象其实都可以用相似的偏微分方程表达。正如常微分方程常被用于对一维动力系统进行建模一样，偏微分方程常被用于对多维系统进行建模。随机偏微分方程延伸了偏微分方程在模拟随机性上的应用。
偏微分方程可以用来描述自然界中各种各样的现象，如声音、热量、静电、电动力学、流体流动、弹性和量子力学等。这些看起来截然不同的物理现象其实都可以用相似的偏微分方程表达。正如常微分方程常被用于对一维动力系统进行建模一样，偏微分方程常被用于对多维系统进行建模。随机偏微分方程延伸了偏微分方程在模拟随机性上的应用。
===非线性微分方程===
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.
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.
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。