微分方程

此词条暂由Yuling翻译，未经人工整理和审校，带来阅读不便，请见谅。

文件:Elmer-pump-heatequation.png

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.

通过求解热方程，建立了泵壳内传热的可视化模型。热量在外壳内部产生并在边界处冷却，从而提供稳定的温度分布。

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，微分方程 Differential Equation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。

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。

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.

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.

通常情况下，当解的封闭形式不存在时，可以用计算机进行近似计算方程的解。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，与此同时也已经发展了许多数值方法来计算给定精度下微分方程的解。

History

历史

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:

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首次出现的。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:

[math]\displaystyle{ \lt math\gt 数学 \begin{align} \begin{align} Begin { align } & \frac {dy}{dx} = f(x) \\[5pt] & \frac {dy}{dx} = f(x) \\[5pt] (x)[5 pt ] & \frac {dy}{dx} = f(x,y) \\[5pt] & \frac {dy}{dx} = f(x,y) \\[5pt] (x，y)[5 pt ] & x_1 \frac {\partial y}{\partial x_1} + x_2 \frac {\partial y}{\partial x_2} = y & x_1 \frac {\partial y}{\partial x_1} + x_2 \frac {\partial y}{\partial x_2} = y & x 1 frac 部分 x 1} + x 2 frac 部分 y 2} y \end{align} \end{align} End { align } }[/math]

</math>

数学

In all these cases, y is an unknown function of x (or of [math]\displaystyle{ x_1 }[/math] and [math]\displaystyle{ x_2 }[/math]), and f is a given function.

In all these cases, is an unknown function of (or of [math]\displaystyle{ x_1 }[/math] and [math]\displaystyle{ x_2 }[/math]), and is a given function.

在这些情况中，y是自变量x(或者是[math]\displaystyle{ x_1 }[/math] and [math]\displaystyle{ x_2 }[/math])的未知函数，并且f是一个给定的函数。

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.

他利用无穷级数解这些例子和其他例子，并讨论了解的非唯一性。

Jacob Bernoulli proposed the Bernoulli differential equation in 1695. 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

雅各布·伯努利在1695年提出了伯努利微分方程。这种方程是常微分方程 Ordinary Differential Equation的一种形式，

[math]\displaystyle{ y'+ P(x)y = Q(x)y^n\, }[/math]

[math]\displaystyle{ y'+ P(x)y = Q(x)y^n\, }[/math]

[math]\displaystyle{ y'+ P(x)y = Q(x)y^n\, }[/math]

for which the following year Leibniz obtained solutions by simplifying it.

for which the following year Leibniz obtained solutions by simplifying it.

莱布尼茨并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. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.

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，弦振动的问题，比如乐器的弦，是由让·勒朗·达朗贝尔，欧拉，丹尼尔·伯努利和约瑟夫·路易斯·拉格朗日研究的。1746年，达朗贝尔发现了一维波动方程，10年之内，欧拉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。这是一个与起点无关的求解曲线的问题，其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，从而导致了拉格朗日力学的形成。 ==Yuling（讨论） independent of the starting point 这里翻译不太好。

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.

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工作，其中他以牛顿的冷却定律为基础进行推理，即两个相邻分子之间的热流与它们之间极小的温差成正比。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可以学到这个偏微分方程。

Example

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，物体的运动是由其位置和速度随时间变化而变化的。牛顿定律允许动态地表示这些变量(给定位置、速度、加速度和作用在物体上的各种力) ，作为物体作为时间函数的未知位置的微分方程。

In some cases, this differential equation (called an equation of motion) may be solved explicitly.

In some cases, this differential equation (called an equation of motion) may be solved explicitly.

在某些情况下，这个微分方程(称为运动方程)可以明确地求解。

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减速度。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，是其速度的导数，取决于速度(而速度取决于时间)。找到速度作为时间的函数需要解决一个微分方程问题并验证它的有效性。

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，这些类型的微分方程可以帮助指导解决方案的选择。常用的区别包括方程是否为: 普通 / 偏微分方程、线性 / 非线性方程和齐次 / 异质方程。这份清单远非详尽无遗; 在特定的情况下，微分方程还有许多其他的性质和子类非常有用。

Ordinary differential equations

Main articles: Ordinary differential equation and Linear differential equation

An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus 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 ), which, therefore, depends on . Thus 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.

常微分方程是一个方程，包含一个实变量或复变量的未知函数，它的导数，和一些给定的函数。未知函数通常由一个变量(通常表示)表示，因此这个变量依赖于。因此通常被称为方程式的自变量。“普通”一词与偏微分方程一词形成对比，后者可能涉及一个以上的独立变量。

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.

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.

线性微分方程是在未知函数及其导数中线性的微分方程。他们的理论发展得很好，在许多情况下可以用积分来表示他们的解。

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).

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(见完整函数)。

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.

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.

一般来说，微分方程的解不能用解析解表示，数值方法通常用于在计算机上求解微分方程。

Partial differential equations

Main article: Partial differential equation

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.

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.

偏微分方程函数是一种包含未知多变量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。)偏微分方程用于描述涉及多个变量函数的问题，或者以封闭形式求解，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, 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.

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、弹性或量子力学。这些看起来截然不同的物理现象可以用偏微分方程类似地形式化。正如常微分方程经常对一维动力系统进行建模一样，偏微分方程经常对多维系统进行建模。随机偏微分方程推广了随机性建模的偏微分方程。

Non-linear differential equations

Main article: 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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参考文献

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，对于小幅度振荡是有效的(见下文)。

模板:Anchor Equation order

Differential equations are described by their order, determined by the term with the highest derivatives. An equation containing only first derivatives is a first-order differential equation, an equation containing the second derivative is a second-order differential equation, and so on. Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin film equation, which is a fourth order partial differential equation.

Differential equations are described by their order, determined by the term with the highest derivatives. An equation containing only first derivatives is a first-order differential equation, an equation containing the second derivative is a second-order differential equation, and so on. Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin film equation, which is a fourth order partial differential equation.

微分方程是用它们的阶来描述的，由导数最高的项来确定。只含有一阶导数的方程是一阶微分方程，含有二阶导数的方程是二阶微分方程，等等。描述自然现象的微分方程几乎总是只有一阶和二阶导数，但也有一些例外，例如薄膜方程，它是一个四阶偏微分方程。

Examples

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.

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.

在第一组例子中，u 是 x 和 c 的未知函数，它们是应该已知的常数。常微分方程和偏微分方程的两种广义分类包括区分线性和非线性微分方程，以及区分齐次微分方程和非齐次微分方程。

• Heterogeneous first-order linear constant coefficient ordinary differential equation:

[math]\displaystyle{ \frac{du}{dx} = cu+x^2. }[/math]

[math]\displaystyle{  \frac{du}{dx} = cu+x^2.  }[/math]

数学[ frac }{ dx } cu + x ^ 2。数学

• Homogeneous second-order linear ordinary differential equation:

[math]\displaystyle{ \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. }[/math]

[math]\displaystyle{  \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0.  }[/math]

数学框架 ^ 2u }{ dx ^ 2}-框架{ dx } + u 0。数学

• Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:

[math]\displaystyle{ \frac{d^2u}{dx^2} + \omega^2u = 0. }[/math]

[math]\displaystyle{  \frac{d^2u}{dx^2} + \omega^2u = 0.  }[/math]

数学框架 ^ 2u }{ dx ^ 2} + omega ^ 2u 0。数学

• Heterogeneous first-order nonlinear ordinary differential equation:

[math]\displaystyle{ \frac{du}{dx} = u^2 + 4. }[/math]

[math]\displaystyle{  \frac{du}{dx} = u^2 + 4.  }[/math]

2 + 4.数学

• Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length L:

[math]\displaystyle{ L\frac{d^2u}{dx^2} + g\sin u = 0. }[/math]

[math]\displaystyle{  L\frac{d^2u}{dx^2} + g\sin u = 0.  }[/math]

数学 l frac { d ^ 2 u }{ dx ^ 2} + g sin u 0。数学

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.

在下一组例子中，未知函数 u 依赖于两个变量 x 和 t 或 x 和 y。

• Homogeneous first-order linear partial differential equation:

[math]\displaystyle{ \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0. }[/math]

[math]\displaystyle{  \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0.  }[/math]

数学部分 u } + t frac 部分 u }0。数学

• Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:

[math]\displaystyle{ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. }[/math]

[math]\displaystyle{  \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0.  }[/math]

部分 x ^ 2} + 部分 y ^ 2}0。数学

• Homogeneous third-order non-linear partial differential equation :

[math]\displaystyle{ \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. }[/math]

[math]\displaystyle{  \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}.  }[/math]

数学 frac { partial u }6u frac { partial u }-frac { partial ^ 3 u }。数学

Existence of solutions

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.

解微分方程不同于解代数方程。不仅他们的解决方案往往不清楚，而且解决方案是否独一无二或是否存在也是值得关注的问题。

For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point [math]\displaystyle{ (a,b) }[/math] in the xy-plane, define some rectangular region [math]\displaystyle{ Z }[/math], such that [math]\displaystyle{ Z = [l,m]\times[n,p] }[/math] and [math]\displaystyle{ (a,b) }[/math] is in the interior of [math]\displaystyle{ Z }[/math]. If we are given a differential equation [math]\displaystyle{ \frac{dy}{dx} = g(x,y) }[/math] and the condition that [math]\displaystyle{ y=b }[/math] when [math]\displaystyle{ x=a }[/math], then there is locally a solution to this problem if [math]\displaystyle{ g(x,y) }[/math] and [math]\displaystyle{ \frac{\partial g}{\partial x} }[/math] are both continuous on [math]\displaystyle{ Z }[/math]. This solution exists on some interval with its center at [math]\displaystyle{ 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]\displaystyle{ (a,b) }[/math] in the xy-plane, define some rectangular region [math]\displaystyle{ Z }[/math], such that [math]\displaystyle{ Z = [l,m]\times[n,p] }[/math] and [math]\displaystyle{ (a,b) }[/math] is in the interior of [math]\displaystyle{ Z }[/math]. If we are given a differential equation [math]\displaystyle{ \frac{dy}{dx} = g(x,y) }[/math] and the condition that [math]\displaystyle{ y=b }[/math] when [math]\displaystyle{ x=a }[/math], then there is locally a solution to this problem if [math]\displaystyle{ g(x,y) }[/math] and [math]\displaystyle{ \frac{\partial g}{\partial x} }[/math] are both continuous on [math]\displaystyle{ Z }[/math]. This solution exists on some interval with its center at [math]\displaystyle{ 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存在解的情况。给定 xy 平面上的任意点数学(a，b) / math，定义一些矩形区域数学 z / math，比如说，math z [ l，m ] times [ n，p ] / math (a，b) / math 是在 math z / math 的内部。如果我们给出一个微分方程数学问题 g (x，y) / math 和数学问题 y b / math 当数学问题 x a / math 时的条件，那么如果数学问题 g (x，y) / math 和 frac 部分数学问题都是数学问题 z / math 上的连续问题，那么这个问题就有一个局部解。这个解决方案以数学 a / math 为中心，以某个时间间隔存在。解决方案可能不是唯一的。(其他结果请参见常微分方程。)

However, this only helps us with first order initial value problems. 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:

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 阶线性初值问题:

[math]\displaystyle{ \lt math\gt 数学 f_{n}(x)\frac{d^n y}{dx^n} + \cdots + f_{1}(x)\frac{d y}{dx} + f_{0}(x)y = g(x) f_{n}(x)\frac{d^n y}{dx^n} + \cdots + f_{1}(x)\frac{d y}{dx} + f_{0}(x)y = g(x) F { n }(x) frac { d ^ n y }{ dx ^ n } + cdots + f {1}(x) frac { d y } + f {0}(x) y g (x) }[/math]

</math>

数学

such that

such that

这样

[math]\displaystyle{ \lt math\gt 数学 y(x_{0})=y_{0}, y'(x_{0}) = y'_{0}, y''(x_{0}) = y''_{0}, \cdots y(x_{0})=y_{0}, y'(x_{0}) = y'_{0}, y(x_{0}) = y_{0}, \cdots Y (x {0}) y {0} ，y’(x {0}) y’{0} ，y (x {0}) y {0} ， cdots }[/math]

</math>

数学

For any nonzero [math]\displaystyle{ f_{n}(x) }[/math], if [math]\displaystyle{ \{f_{0},f_{1},\cdots\} }[/math] and [math]\displaystyle{ g }[/math] are continuous on some interval containing [math]\displaystyle{ x_{0} }[/math], [math]\displaystyle{ y }[/math] is unique and exists.

For any nonzero [math]\displaystyle{ f_{n}(x) }[/math], if [math]\displaystyle{ \{f_{0},f_{1},\cdots\} }[/math] and [math]\displaystyle{ g }[/math] are continuous on some interval containing [math]\displaystyle{ x_{0} }[/math], [math]\displaystyle{ y }[/math] is unique and exists.

对于任意非零的数学 f { n }(x) / math，如果数学 f {0}、 f {1}、 cdots / math 和数学 g / math 在某个区间上连续，且包含数学 x {0} / math，则数学 y / math 是唯一存在的。

Related concepts

• 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.

• 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.

• 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.

• A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.

Connection to difference equations

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.

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，坐标系只假定离散值，其关系涉及未知函数或函数的值以及坐标系附近的值。许多计算微分方程数值解或研究微分方程性质的方法涉及到用相应差分方程的解逼近微分方程的解。

Applications

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.

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.

微分方程的研究是一个广泛的领域，在纯粹和应用数学，物理和工程。所有这些学科都与各种类型的微分方程的性质有关。纯数学关注解的存在性和唯一性，而应用数学则强调解的逼近方法的严格性。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。微分方程，例如那些用来解决实际问题的微分方程，可能不一定是直接可解的，例如。没有封闭形式的解。相反，解可以用数值方法来近似。

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结果的应用科学一起发展起来的。然而，不同的问题，有时起源于相当不同的科学领域，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 Joseph Fourier 提出的，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。

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.

在不同的科学领域得到一个名称的微分方程的数量是这个主题的重要性的见证。参见已命名的微分方程列表。

See also

• Complex differential equation

• Exact differential equation

• Functional differential equation

• Initial condition

• Integral equations

• Numerical methods for ordinary differential equations

• Numerical methods for partial differential equations

• Picard–Lindelöf theorem on existence and uniqueness of solutions

• Recurrence relation, also known as 'difference equation'

• Abstract differential equation

• System of differential equations

References

Further reading

• Abbott, P.; Neill, H. (2003). Teach Yourself Calculus. pp. 266–277. 

• Blanchard, P.; Devaney, R. L.; Hall, G. R. (2006). Differential Equations. Thompson. 

• Boyce, W.; DiPrima, R.; Meade, D. (2017). Elementary Differential Equations and Boundary Value Problems. Wiley. 

• Coddington, E. A.; Levinson, N. (1955). Theory of Ordinary Differential Equations. McGraw-Hill. https://archive.org/details/theoryofordinary00codd. 

• Ince, E. L. (1956). Ordinary Differential Equations. Dover. 

• Johnson, W. (1913). A Treatise on Ordinary and Partial Differential Equations. John Wiley and Sons. http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001.  In University of Michigan Historical Math Collection

• Polyanin, A. D.; Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations (2nd ed.). Boca Raton: Chapman & Hall/CRC Press. ISBN 1-58488-297-2. 

• Porter, R. I. (1978). "XIX Differential Equations". Further Elementary Analysis. 

• Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. http://www.mat.univie.ac.at/~gerald/ftp/book-ode/. 

• Daniel Zwillinger (12 May 2014). Handbook of Differential Equations. Elsevier Science. ISBN 978-1-4832-6396-0. https://books.google.com/?id=n7TiBQAAQBAJ&printsec=frontcover&dq=%22Handbook+of+Differential+Equations%22#v=onepage&q=%22Handbook%20of%20Differential%20Equations%22&f=false. 

External links

• 模板:Commonscatinline

• Lectures on Differential Equations MIT Open CourseWare Videos

• Online Notes / Differential Equations Paul Dawkins, Lamar University

• Differential Equations, S.O.S. Mathematics

• Introduction to modeling via differential equations Introduction to modeling by means of differential equations, with critical remarks.

• Mathematical Assistant on Web Symbolic ODE tool, using Maxima

• Exact Solutions of Ordinary Differential Equations

• Collection of ODE and DAE models of physical systems MATLAB models

• Notes on Diffy Qs: Differential Equations for Engineers An introductory textbook on differential equations by Jiri Lebl of UIUC

• Khan Academy Video playlist on differential equations Topics covered in a first year course in differential equations.

• MathDiscuss Video playlist on differential equations

This page was moved from wikipedia:en:Differential equation. Its edit history can be viewed at 微分方程/edithistory