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第1行： − 此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。+ 第7行：
第7行： − + − 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+ 第15行：
第15行： − 在数学中，微分方程是一个关联一个或多个函数及其导数的方程。在应用程序中，函数通常代表物理量，导数代表变化率，微分方程定义了两者之间的关系。这种关系是常见的，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起着突出的作用。+ 第23行：
第23行： − + 第31行：
第31行： − 通常情况下，当解的解析解不存在时，解可以用计算机进行近似计算。动力系统理论强调对由微分方程描述的系统进行定性分析，而已经发展了许多数值方法来确定给定精度的解。+ − + 第93行：
第93行： − 在所有这些情况下，都是一个未知函数(或者是数学 x1 / math 和数学 x2 / math) ，并且是一个给定的函数。+ 第101行：
第101行： − 他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。+ 第109行：
第109行： − 雅各布 · 伯努利在1695年提出了伯努利微分方程。这是这种形式的一个常微分方程+ 第117行：
第117行： − 数学 y’ + p (x) y q (x) y ^ n ，/ math+ 第125行：
第125行： − 第二年，莱布尼茨通过简化得到了解。+ 第133行：
第133行： − 历史上，振动弦的问题，比如乐器的问题，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，d’ alembert 发现了一维波动方程，10年之内，Euler 发现了三维波动方程。+ 第141行：
第141行： − + − +
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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。
{{short description|Mathematical equation involving derivatives of an unknown function}}
{{short description|Mathematical equation involving derivatives of an unknown function}}
[[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.]]
[[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.]]
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 [[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 [[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.
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。
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，一个给定微分方程的解的许多性质可能不需要精确计算就可以确定。
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==
==History==
历史
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, 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> and <math>x_2</math>)的未知函数，并且{{mvar|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'''<font color="#ff8000">常微分方程 Ordinary Differential Equation</font><font>'''的一种形式，
<math>y'+ P(x)y = Q(x)y^n\,</math>
<math>y'+ P(x)y = Q(x)y^n\,</math>
: <math>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兰奇在18世纪50年代结合他们对等角线问题的研究而发明的。这是一个确定曲线的问题，其中一个加权的粒子将下降到一个固定的点在一个固定的时间量，独立于起点。拉格兰奇在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兰奇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。
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。
==[[用户:Yuling|Yuling]]（[[用户讨论:Yuling|讨论]]） independent of the starting point 这里翻译不太好。