differential equation

简明释义

微分方程

英英释义

例句

1.Engineers use differential equations (微分方程) to model the behavior of electrical circuits.

工程师使用differential equations (微分方程)来模拟电路的行为。

2.To solve the heat distribution problem, we need to set up a differential equation (微分方程) based on Fourier's law.

要解决热分布问题，我们需要根据傅里叶定律建立一个differential equation (微分方程)。

3.In physics, many laws are expressed as differential equations (微分方程), such as Newton's second law.

在物理学中，许多定律以differential equations (微分方程)的形式表达，例如牛顿第二定律。

4.The motion of a pendulum can be described by a second-order differential equation (微分方程).

摆的运动可以用一个二阶differential equation (微分方程)来描述。

5.The population growth can often be modeled by a logistic differential equation (微分方程).

人口增长通常可以通过逻辑斯蒂differential equation (微分方程)建模。

作文

Differential equations are a fundamental part of mathematics that describe how a quantity changes in relation to another quantity. They are used extensively in various fields such as physics, engineering, biology, and economics. A differential equation (微分方程) involves derivatives, which represent rates of change. Understanding these equations is crucial for modeling real-world phenomena, from the motion of planets to the spread of diseases.To grasp the concept of differential equations (微分方程), it is essential to start with the basics of calculus, particularly the notion of derivatives. A derivative measures how a function changes as its input changes. For example, if we consider the position of an object over time, its velocity is the derivative of its position with respect to time. Similarly, acceleration is the derivative of velocity. Thus, differential equations (微分方程) can be seen as mathematical expressions that relate these rates of change.There are two main types of differential equations (微分方程): ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve functions of a single variable and their derivatives, while PDEs involve multiple variables. For instance, the equation describing the motion of a pendulum is an ODE, whereas the heat equation, which describes how heat diffuses through a medium, is a PDE.Solving differential equations (微分方程) is often challenging, and various techniques have been developed to find solutions. Some common methods include separation of variables, integrating factors, and numerical approaches. Each method has its applications depending on the nature of the equation and the context in which it arises.One of the most famous differential equations (微分方程) is Newton's second law of motion, which can be expressed as F = ma, where F is the force applied to an object, m is its mass, and a is its acceleration. This relationship can be rewritten as a differential equation (微分方程) that describes how the velocity of an object changes over time when subjected to a specific force. This example illustrates how differential equations (微分方程) can model physical systems and predict their behavior under various conditions.Another significant application of differential equations (微分方程) is in population dynamics, where they are used to model the growth of populations over time. The logistic growth model, for example, is a differential equation (微分方程) that describes how a population grows rapidly at first but then slows down as it approaches the carrying capacity of its environment. This model helps biologists understand the factors that influence population size and sustainability.In conclusion, differential equations (微分方程) are powerful mathematical tools that allow us to model and analyze a wide range of dynamic systems. Their versatility and applicability in various scientific fields make them an essential area of study for anyone interested in understanding the complexities of the world around us. By mastering the principles of differential equations (微分方程), one can gain valuable insights into the behavior of systems, leading to advancements in technology, medicine, and environmental science. As we continue to explore the intricacies of these equations, we unlock new possibilities for innovation and discovery in our ever-evolving world.

微分方程是数学的一个基本部分，描述了一个量如何相对于另一个量变化。它们在物理学、工程学、生物学和经济学等各个领域被广泛使用。differential equation（微分方程）涉及导数，这些导数表示变化率。理解这些方程对建模现实世界现象至关重要，从行星的运动到疾病的传播。要掌握differential equations（微分方程）的概念，首先必须了解微积分的基础知识，特别是导数的概念。导数衡量一个函数在输入变化时的变化。例如，如果我们考虑一个物体随时间的位置，其速度就是位置相对于时间的导数。同样，加速度是速度的导数。因此，differential equations（微分方程）可以看作是将这些变化率联系起来的数学表达式。differential equations（微分方程）主要有两种类型：常微分方程（ODE）和偏微分方程（PDE）。ODE涉及单变量的函数及其导数，而PDE则涉及多个变量。例如，描述摆动运动的方程是ODE，而描述热量如何在介质中扩散的热方程是PDE。求解differential equations（微分方程）通常具有挑战性，因此开发了多种技术来寻找解决方案。一些常见的方法包括变量分离、积分因子和数值方法。每种方法都有其应用，具体取决于方程的性质和出现的背景。其中一个最著名的differential equations（微分方程）是牛顿第二运动定律，可以表示为F = ma，其中F是施加在物体上的力，m是其质量，a是其加速度。这个关系可以重写为描述在特定力作用下，物体的速度如何随时间变化的differential equation（微分方程）。这个例子说明了differential equations（微分方程）如何建模物理系统并预测它们在各种条件下的行为。differential equations（微分方程）另一个重要的应用是在种群动态中，用于模型化种群随时间的增长。例如，逻辑增长模型就是一个描述种群最初快速增长，但随着接近环境的承载能力而减缓的differential equation（微分方程）。该模型帮助生物学家理解影响种群大小和可持续性的因素。总之，differential equations（微分方程）是强大的数学工具，使我们能够建模和分析广泛的动态系统。它们的多功能性和在各个科学领域的适用性使它们成为任何想要理解周围世界复杂性的人必不可少的研究领域。通过掌握differential equations（微分方程）的原理，人们可以深入洞察系统的行为，从而推动技术、医学和环境科学的发展。随着我们继续探索这些方程的复杂性，我们为我们不断发展的世界中的创新和发现打开了新的可能性。