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第392行：
第392行： − + 第400行：
第400行： − 微分方程的研究涉及很多领域，例如理论数学，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。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，比如它们没有封闭形式的解。但取而代之的是，我们可以用数值方法来近似方程的解。+ 第408行：
第408行： − 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。+
→Applications
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，坐标系中只假定存在离散值，计算中会涉及到未知函数或已知函数的值以及坐标附近的值。许多计算微分方程数值解或研究微分方程性质的方法都会涉及到通过相应差分方程的解来逼近微分方程的解这一过程。
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，坐标系中只假定存在离散值，计算中会涉及到未知函数或已知函数的值以及坐标附近的值。许多计算微分方程数值解或研究微分方程性质的方法都会涉及到通过相应差分方程的解来逼近微分方程的解这一过程。
==Applications==
==Applications 应用==
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.
微分方程的研究可以应用于许多领域，如理论数学、应用数学、物理学和工程学，它们都与各种类型的微分方程的性质有关。理论数学关注解的存在性和唯一性，而应用数学则强调求解方法的严格准确性。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。用于解决实际问题的微分方程，不一定是直接可解的，如可能不存在闭式解。但我们可以用数值方法来近似得到方程的解。
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都可以用微分方程来表示。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。微分方程理论最初是与其起源并得到应用的科学一起发展起来的89。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。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，光和声在大气中的传播，或是池塘表面的水波的传播。所有这些过程都可以用相同的二阶偏微分方程来描述，即波动方程。我们把光和声音想象成与水波相似的形式。由约瑟夫·傅里叶提出的热传导的理论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，虽然看上去形式不同，但都可以同一个方程来描述;。例如，金融学中的布莱克-斯科尔斯方程就与热方程有关。