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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。

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}}

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.

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，而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。

在数学上，'''<font color="#ff8000">微分方程 Differential Equation</font><font>'''是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，而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得到了广泛的应用。

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.