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第44行：
第44行： − 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:+ 第60行：
第60行： − 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。+ 第68行：
第68行： − 他利用无穷级数求解这些例子和其他例子，并讨论了解的非唯一性。+ 第87行：
第87行： − 莱布尼茨并于第二年将方程简化从而得到了方程的解。+ 第95行：
第95行： − 历史上，弦振动问题——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。+ 第103行：
第103行： − + 第111行：
第111行： − 1822年，傅立叶在《热的分析理论》中发表了他关于热流的研究工作，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需要学习这类偏微分方程。+
→History
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:
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.
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。
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.
他利用无穷级数来求解这些以及其他例子，并讨论了解的非唯一性。
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，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。
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。
==[[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 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==
==Example==