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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.

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.

在数学中，'''<font color="#ff8000">微分方程 Differential Equation</font><font>'''是一个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，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有着突出的作用。

在数学中，'''<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有着突出的作用。

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.  

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用显式公式求解；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。

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}} 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, is an unknown function of  (or of <math>x_1</math> and <math>x_2</math>), and  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.

在这些情况中，{{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。

在这些情况中，{{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。

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。这是一个与起点无关的求解曲线的问题，问题中一个加权的粒子将在一个固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，并将其发送给欧拉。两者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。

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求解曲线问题，问题中一个加权的粒子将在一个固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，并将其发送给欧拉。两者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。

==[[用户:Yuling|Yuling]]（[[用户讨论:Yuling|讨论]]） independent of the starting point 这里翻译不太好。

==[[用户:Yuling|Yuling]]（[[用户讨论:Yuling|讨论]]） independent of the starting point 这里翻译不太好。

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 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.

在经典力学中，物体的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) ，并给出了求解时间的函数——物体未知位置——的微分方程。

在经典力学中，物体的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。

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.

使用微分方程模拟现实世界问题的一个例子是仅考虑重力和空气阻力确定球在空中落下的速度。球对地面的加速度是由于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它的有效性。

使用微分方程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由于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。

==Types==

==Types==

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.

微分方程可分为几种类型。除了描述方程本身的性质之外，微分方程的类型有助于指导选择何种微分方程作为解决方案。常见的区别包括方程是否为: 常微分/偏微分方程、线性/非线性方程和齐次/非齐次方程。这份清单远非详尽无遗; 在特定的情况下，微分方程还有许多非常有用的其他性质和子类。

微分方程可分为几种类型。除了描述方程本身的性质之外，微分方程的类型有助于指导选择何种解决方案。常见的区别包括方程是否为: 常微分/偏微分方程、线性/非线性方程和齐次/非齐次方程。这份清单还很长，微分方程还有许多在特定的情况下非常有用的其他性质和子类。

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.

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.

常微分方程中，包含有：只含有一个实变量或复变量的未知函数及其导数以及一些已知的函数。未知函数通常由一个变量(通常由 {{mvar|y}} 表示)表示，因此这个变量依赖于 {{mvar|x}} 。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，后者可能涉及一个以上的独立变量。

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。未知函数通常由一个变量(通常由 {{mvar|y}} 表示)表示，因此这个变量依赖于 {{mvar|x}} 。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。

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.

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===

===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).

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.)

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> ，定义一些矩形区域 <math>Z</math> ，比如说，<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> 和条件：当<math>x=a</math>时<math>y=b</math>，那么如果<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。这个解决方案以 <math>a</math> 为中心的某些区间上存在。方程的解可能不是唯一的。(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> ，定义一些矩形区域 <math>Z</math> ，比如说，<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，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。)

* 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.

* 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.

*An [[integro-differential equation]] (IDE) is an equation that combines aspects of a differential equation and an [[integral equation]].

*An [[integro-differential equation]] (IDE) is an equation that combines aspects of a differential equation and an [[integral 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.

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可以表述为微分方程。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。微分方程的数学理论最初是与方程的起源和方程解的搜索一起发展起来。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。每当这种情况发生时，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，考虑光和声在大气中的传播，以及池塘表面的波的传播。所有这些都可以用相同的二阶偏微分方程来描述，即波动方程，它允许我们把光和声音想象成波的形式，很像水中熟悉的波。热传导的理论是由约瑟夫.傅里叶提出的，这一过程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，虽然看起来不同，却用同一个方程来描述; 例如，金融学中的布莱克-斯科尔斯方程就与热方程有关。

许多物理和化学的基本定律可以表述为微分方程。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。微分方程的数学理论最初是与微分方程起源于和得到应用的科学一起发展起来。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。每当这种情况发生时，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，考虑光和声在大气中的传播，以及池塘表面的波的传播。所有这些都可以用相同的二阶偏微分方程来描述，即波动方程，它允许我们把光和声音想象成波的形式，很像水中相似的波。热传导的理论是由约瑟夫.傅里叶提出的，这一过程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，虽然看起来不同，却用同一个方程来描述; 例如，金融学中的布莱克-斯科尔斯方程就与热方程有关。