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第99行： − − − − In the first group of examples ''u'' is an unknown function of ''x'', and ''c'' and ''ω'' are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between ''[[linear differential equation|linear]]'' and ''nonlinear'' differential equations, and between [[homogeneous differential equation|''homogeneous'' differential equation]]s and ''heterogeneous'' ones. − − In the first group of examples u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones. − + − − 非齐次一阶常系数常微分方程 第118行：
第110行： + − * Homogeneous second-order linear ordinary differential equation: − 齐次二阶线性常微分方程 − + − 用于描述简谐振动的齐次二阶常系数常系数微分方程 − + − 非齐次一阶非线性常微分方程 − + − − 用于描述摆长为L的钟摆运动的二阶非线性（因正弦函数产生）常微分方程 第145行：
第132行： − − In the next group of examples, the unknown function ''u'' depends on two variables ''x'' and ''t'' or ''x'' and ''y''. − − In the next group of examples, the unknown function u depends on two variables x and t or x and y. 第154行：
第137行： − + − 齐次一阶线性偏微分方程 第163行：
第145行： − + − 齐次二阶线性常系数椭圆形偏微分方程，也称为拉普拉斯方程 第170行：
第151行： − + − − − 齐次三阶非线性偏微分方程 − + − 解的存在性 − − Solving differential equations is not like solving [[algebraic equations]]. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest. − Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest. − − − For first order initial value problems, the [[Peano existence theorem]] gives one set of circumstances in which a solution exists. Given any point <math>(a,b)</math> in the xy-plane, define some rectangular region <math>Z</math>, such that <math>Z = [l,m]\times[n,p]</math> and <math>(a,b)</math> is in the interior of <math>Z</math>. If we are given a differential equation <math>\frac{dy}{dx} = g(x,y)</math> and the condition that <math>y=b</math> when <math>x=a</math>, then there is locally a solution to this problem if <math>g(x,y)</math> and <math>\frac{\partial g}{\partial x}</math> are both continuous on <math>Z</math>. This solution exists on some interval with its center at <math>a</math>. The solution may not be unique. (See [[Ordinary differential equation]] for other results.) − − For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point <math>(a,b)</math> in the xy-plane, define some rectangular region <math>Z</math>, such that <math>Z = [l,m]\times[n,p]</math> and <math>(a,b)</math> is in the interior of <math>Z</math>. If we are given a differential equation <math>\frac{dy}{dx} = g(x,y)</math> and the condition that <math>y=b</math> when <math>x=a</math>, then there is locally a solution to this problem if <math>g(x,y)</math> and <math>\frac{\partial g}{\partial x}</math> are both continuous on <math>Z</math>. This solution exists on some interval with its center at <math>a</math>. The solution may not be unique. (See Ordinary differential equation for other results.) − − − However, this only helps us with first order [[initial value problem]]s. Suppose we had a linear initial value problem of the nth order: − − However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order: 第210行：
第174行： − such that − − such that 第221行：
第182行： − For any nonzero <math>f_{n}(x)</math>, if <math>\{f_{0},f_{1},\cdots\}</math> and <math>g</math> are continuous on some interval containing <math>x_{0}</math>, <math>y</math> is unique and exists.<ref>{{cite book|last1=Zill|first1=Dennis G.|title=A First Course in Differential Equations|publisher=Brooks/Cole|isbn=0-534-37388-7|edition=5th|year=2001}}</ref>+ − − For any nonzero <math>f_{n}(x)</math>, if <math>\{f_{0},f_{1},\cdots\}</math> and <math>g</math> are continuous on some interval containing <math>x_{0}</math>, <math>y</math> is unique and exists. − − 对于任意非零 <math>f_{n}(x)</math> ，如果<math>\{f_{0},f_{1},\cdots\}</math> 和 <math>g</math>在某个包含<math>x_{0}</math>的区间上连续，则<math>y</math>是存在且唯一的。 − − ==Related concepts== − 相关概念 − − * A [[delay differential equation]] (DDE) is an equation for a function of a single variable, usually called '''time''', in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.+ − 延迟微分方程（DDE）是一元函数的方程，变量通常为时间，其中函数在一定时间点的微分会被较早时间点的函数值表达。 − *An [[integro-differential equation]] (IDE) is an equation that combines aspects of a differential equation and an [[integral equation]]. − 积分微分方程（IDE）结合了微分方程和积分方程。 − * A [[stochastic differential equation]] (SDE) is an equation in which the unknown quantity is a [[stochastic process]] and the equation involves some known stochastic processes, for example, the [[Wiener process]] in the case of diffusion equations. − 随机微分方程（SDE）中的未知量处于随机过程，并且涉及一些已知的随机过程，例如，扩散方程中的维纳过程。 − + − 随机偏微分方程（SPDE）是一种含空间和时间噪声过程的广义随机微分方程，它通常应用于量子场论以及统计力学中。 − + − 微分代数方程（DAE）是一种含微分和代数项的微分方程，通常以隐式形式给出。 − ==Connection to difference equations==+ − 与差分方程之间的联系 − {{See also|Time scale calculus}} + + − The theory of differential equations is closely related to the theory of [[difference equations]], in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation.+ − The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation. − + − The study of differential equations is a wide field in [[pure mathematics|pure]] and [[applied mathematics]], [[physics]], and [[engineering]]. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have [[closed-form expression|closed form]] solutions. Instead, solutions can be approximated using [[Numerical ordinary differential equations|numerical methods]]. − − The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods. 第270行：
第212行： − Many fundamental laws of [[physics]] and [[chemistry]] can be formulated as differential equations. In [[biology]] and [[economics]], differential equations are used to [[mathematical modelling|model]] the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order [[partial differential equation]], the [[wave equation]], which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by [[Joseph Fourier]], is governed by another second-order partial differential equation, the [[heat equation]]. It turns out that many [[diffusion]] processes, while seemingly different, are described by the same equation; the [[Black–Scholes]] equation in finance is, for instance, related to the heat equation.+ − Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation. − − 许多物理和化学的基本定律都可以用微分方程来表示。在生物学和经济学中，微分方程被用来模拟复杂系统的行为。微分方程理论最初是与其起源并得到应用的科学一起发展起来的89。然而，有时完全不同的科学领域，却可能产生相同的微分方程。当这种情况发生时，方程后面的数学理论可以被看作是不同现象背后的统一原则。例如，光和声在大气中的传播，或是池塘表面的水波的传播。所有这些过程都可以用相同的二阶偏微分方程来描述，即波动方程。我们把光和声音想象成与水波相似的形式。由约瑟夫·傅里叶提出的热传导的理论由另一个二阶偏微分方程——热方程所支配。事实证明，许多扩散过程，虽然看上去形式不同，但都可以同一个方程来描述;。例如，金融学中的布莱克-斯科尔斯方程就与热方程有关。 − − − ==[[用户:Yuling|Yuling]]（[[用户讨论:Yuling|讨论]]） "results found application" 翻译为“方程解的搜索”，可能不太准确 − − − The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See [[List of named differential equations]]. − − The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See List of named differential equations. − + − − {{Div col|colwidth=22em}} − + − + − + − − − + − + − + − + − + − + − + − + − − − − − − + − + 第340行：
第262行： − +
无编辑摘要
===示例===
===示例===
在第一组示例中，待求解的''u''是''x''的函数，''c''和''ω''是应该已知的常数。常微分方程和偏微分方程这两种广义分类下还要区分微分方程的线性和非线性，以及区分微分方程的齐次和非齐次。
在第一组示例中，待求解的''u''是''x''的函数，''c''和''ω''是应该已知的常数。常微分方程和偏微分方程这两种广义分类下还要区分微分方程的线性和非线性，以及区分微分方程的齐次和非齐次。
* 非齐次一阶常系数常微分方程:
* Heterogeneous first-order linear constant coefficient ordinary differential equation:
* 齐次二阶线性常微分方程:
:: <math> \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. </math>
:: <math> \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. </math>
* Homogeneous second-order linear constant coefficient ordinary differential equation describing the [[harmonic oscillator]]:
* 用于描述简谐振动的齐次二阶常系数常系数微分方程:
:: <math> \frac{d^2u}{dx^2} + \omega^2u = 0. </math>
:: <math> \frac{d^2u}{dx^2} + \omega^2u = 0. </math>
* Heterogeneous first-order nonlinear ordinary differential equation:
* 非齐次一阶非线性常微分方程:
:: <math> \frac{du}{dx} = u^2 + 4. </math>
:: <math> \frac{du}{dx} = u^2 + 4. </math>
* 用于描述摆长为L的钟摆运动的二阶非线性（因正弦函数产生）常微分方程:
* Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a [[pendulum]] of length ''L'':
在下一组例子中，未知函数''u''依赖于两个变量''x'' 和 ''t''或者''x''和''y''。
在下一组例子中，未知函数''u''依赖于两个变量''x'' 和 ''t''或者''x''和''y''。
* Homogeneous first-order linear partial differential equation:
* 齐次一阶线性偏微分方程:
* Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the [[Laplace equation]]:
*齐次二阶线性常系数椭圆形偏微分方程，也称为拉普拉斯方程:
*齐次三阶非线性偏微分方程:
* Homogeneous third-order non-linear partial differential equation :
:: <math> \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. </math>
:: <math> \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. </math>
==Existence of solutions==
==解的存在性==
解微分方程不同于解代数方程。方程解的情况往往是不确定的，而且解是否唯一或是否存在也是值得关注的问题。
解微分方程不同于解代数方程。方程解的情况往往是不确定的，而且解是否唯一或是否存在也是值得关注的问题。
对于一阶初值问题，皮亚诺存在性定理给出了一组解存在的情况。给定的x-y平面上的任意点 <math>(a,b)</math> ，定义矩形区域 <math>Z</math> ，如，<math>Z = [l,m]\times[n,p]</math> 而且 <math>(a,b)</math> 是 <math>Z</math> 内部一点。如果我们给出一个微分方程 <math>\frac{dy}{dx} = g(x,y)</math> 和当<math>x=a</math>时<math>y=b</math>，如果<math>g(x,y)</math>和<math>\frac{\partial g}{\partial x}</math>在<math>Z</math>上是连续的，那么这个问题就有一个局部解。这个解在以 <math>a</math> 为中心的某些区间上存在，其可能不是唯一的。(其他结果请参见常微分方程。)
对于一阶初值问题，皮亚诺存在性定理给出了一组解存在的情况。给定的x-y平面上的任意点 <math>(a,b)</math> ，定义矩形区域 <math>Z</math> ，如，<math>Z = [l,m]\times[n,p]</math> 而且 <math>(a,b)</math> 是 <math>Z</math> 内部一点。如果我们给出一个微分方程 <math>\frac{dy}{dx} = g(x,y)</math> 和当<math>x=a</math>时<math>y=b</math>，如果<math>g(x,y)</math>和<math>\frac{\partial g}{\partial x}</math>在<math>Z</math>上是连续的，那么这个问题就有一个局部解。这个解在以 <math>a</math> 为中心的某些区间上存在，其可能不是唯一的。(其他结果请参见常微分方程。)
然而，这只能帮助我们解决一阶初始值问题。假设我们有一个n阶线性初始值问题:
然而，这只能帮助我们解决一阶初始值问题。假设我们有一个n阶线性初始值问题:
</math>
</math>
其中有
其中有
对于任意非零 <math>f_{n}(x)</math> ，如果<math>\{f_{0},f_{1},\cdots\}</math> 和 <math>g</math>在某个包含<math>x_{0}</math>的区间上连续，则<math>y</math>是存在且唯一的。<ref>{{cite book|last1=Zill|first1=Dennis G.|title=A First Course in Differential Equations|publisher=Brooks/Cole|isbn=0-534-37388-7|edition=5th|year=2001}}</ref>
==相关概念==
*A [[stochastic partial differential equation]] (SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications in [[quantum field theory]] and [[statistical mechanics]].
*延迟微分方程（DDE）是一元函数的方程，变量通常为时间，其中函数在一定时间点的微分会被较早时间点的函数值表达。
* A [[differential algebraic equation]] (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.
*积分微分方程（IDE）结合了微分方程和积分方程。
*随机微分方程（SDE）中的未知量处于随机过程，并且涉及一些已知的随机过程，例如，扩散方程中的维纳过程。
*随机偏微分方程（SPDE）是一种含空间和时间噪声过程的广义随机微分方程，它通常应用于量子场论以及统计力学中。
* 微分代数方程（DAE）是一种含微分和代数项的微分方程，通常以隐式形式给出。
==与差分方程之间的联系==
微分方程理论与差分方程理论密切相关。在差分方程理论中，坐标系中只假定存在离散值，计算中会涉及到未知函数或已知函数的值以及坐标附近的值。许多求微分方程数值解或研究微分方程性质的方法，都会涉及通过相应差分方程的解来逼近微分方程的解。
微分方程理论与差分方程理论密切相关。在差分方程理论中，坐标系中只假定存在离散值，计算中会涉及到未知函数或已知函数的值以及坐标附近的值。许多求微分方程数值解或研究微分方程性质的方法，都会涉及通过相应差分方程的解来逼近微分方程的解。
==Applications 应用==
== 应用==
微分方程的研究可以应用于许多领域，如理论数学、应用数学、物理学和工程学，它们都与各种类型的微分方程的性质有关。理论数学关注解的存在性和唯一性，而应用数学则强调求解方法的严格准确性。从天体运动到桥梁设计，再到神经元之间的相互作用，微分方程在几乎所有物理、技术或生物过程的建模中都扮演着重要的角色。用于解决实际问题的微分方程，不一定是直接可解的，如可能不存在闭式解。但我们可以用数值方法来近似得到方程的解。
微分方程的研究可以应用于许多领域，如理论数学、应用数学、物理学和工程学，它们都与各种类型的微分方程的性质有关。理论数学关注解的存在性和唯一性，而应用数学则强调求解方法的严格准确性。从天体运动到桥梁设计，再到神经元之间的相互作用，微分方程在几乎所有物理、技术或生物过程的建模中都扮演着重要的角色。用于解决实际问题的微分方程，不一定是直接可解的，如可能不存在闭式解。但我们可以用数值方法来近似得到方程的解。
许多物理和化学的基本定律都可以用微分方程来表示。在生物学和经济学中，微分方程被用来模拟复杂系统的行为。微分方程理论最初是与其起源并得到应用的科学一起发展起来的。然而，有时完全不同的科学领域，却可能产生相同的微分方程。当这种情况发生时，方程后面的数学理论可以被看作是不同现象背后的统一原则。例如，光和声在大气中的传播，或是池塘表面的水波的传播。所有这些过程都可以用相同的二阶偏微分方程来描述，即波动方程。我们把光和声音想象成与水波相似的形式。由约瑟夫·傅里叶提出的热传导的理论由另一个二阶偏微分方程——热方程所支配。事实证明，许多扩散过程，虽然看上去形式不同，但都可以同一个方程来描述;。例如，金融学中的布莱克-斯科尔斯方程就与热方程有关。
事实上，同一类型的微分方程可以应用于不同领域这样的现象屡见不鲜，这足以证明微分方程这一课题的重要性。参见已命名的微分方程列表。
事实上，同一类型的微分方程可以应用于不同领域这样的现象屡见不鲜，这足以证明微分方程这一课题的重要性。参见已命名的微分方程列表。
==See also 另请参见==
==参见==
*[[Complex differential equation]]
*复微分方程
复微分方程
*精确微分方程
*[[Exact differential equation]]
*
精确微分方程
*[[Functional differential equation]]
泛函微分方程
泛函微分方程
*[[Initial condition]]
*
初始条件
初始条件
*[[Integral equations]]
*
积分方程
积分方程
*[[Numerical methods for ordinary differential equations]]
*
求解常微分方程的数值方法
求解常微分方程的数值方法
*[[Numerical methods for partial differential equations]]
*求解偏微分方程的数值方法
求解偏微分方程的数值方法
*关于解的存在性和唯一性的皮卡德–林德洛夫定理
*[[Picard–Lindelöf theorem]] on existence and uniqueness of solutions
*递推关系，也称为差分方程
关于解的存在性和唯一性的皮卡德–林德洛夫定理
*抽象微分方程
*[[Recurrence relation]], also known as 'difference equation'
*微分方程组
递推关系，也称为差分方程
*[[Abstract differential equation]]
抽象微分方程
*[[System of differential equations]]
微分方程组
{{div col end}}
{{div col end}}
==References 参考文献==
==参考文献==
{{reflist|30em}}
{{reflist|30em}}
==Further reading 延伸阅读==
==拓展阅读==
*{{cite book |first=P. |last=Abbott |first2=H. |last2=Neill |title=Teach Yourself Calculus |year=2003 |pages=266–277 }}
*{{cite book |first=P. |last=Abbott |first2=H. |last2=Neill |title=Teach Yourself Calculus |year=2003 |pages=266–277 }}
*{{cite book|author=Daniel Zwillinger|title=Handbook of Differential Equations|url=https://books.google.com/?id=n7TiBQAAQBAJ&printsec=frontcover&dq=%22Handbook+of+Differential+Equations%22#v=onepage&q=%22Handbook%20of%20Differential%20Equations%22&f=false|date=12 May 2014|publisher=Elsevier Science|isbn=978-1-4832-6396-0}}
*{{cite book|author=Daniel Zwillinger|title=Handbook of Differential Equations|url=https://books.google.com/?id=n7TiBQAAQBAJ&printsec=frontcover&dq=%22Handbook+of+Differential+Equations%22#v=onepage&q=%22Handbook%20of%20Differential%20Equations%22&f=false|date=12 May 2014|publisher=Elsevier Science|isbn=978-1-4832-6396-0}}
==External links 外部链接==
==外部链接==
{{wikiquote}}
{{wikiquote}}