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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。
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。
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.
他利用无穷级数求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
雅可比·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，
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。
==[[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。