differential equation

简明释义

微分方程

英英释义

例句

1.In physics, the motion of a falling object can be described by a differential equation (微分方程) that accounts for gravitational force.

在物理学中，掉落物体的运动可以用一个考虑重力的differential equation (微分方程) 来描述。

2.To model the population growth of a species, biologists often use a differential equation (微分方程) that incorporates birth and death rates.

为了模拟一个物种的种群增长，生物学家通常使用包含出生率和死亡率的differential equation (微分方程)。

3.Engineers use differential equations (微分方程) to analyze the behavior of electrical circuits over time.

工程师使用differential equations (微分方程) 来分析电路随时间的行为。

4.Economists might employ a differential equation (微分方程) to predict changes in market trends over time.

经济学家可能会使用一个differential equation (微分方程) 来预测市场趋势随时间的变化。

5.The heat distribution in a solid object can be modeled using a partial differential equation (偏微分方程).

固体物体中的热分布可以使用一个偏differential equation (偏微分方程) 来建模。

作文

Differential equations are a fundamental concept in mathematics and engineering, serving as a crucial tool for modeling various phenomena in the natural world. A differential equation (微分方程) is an equation that relates a function with its derivatives. These equations arise in numerous fields, including physics, biology, economics, and even social sciences, illustrating their versatility and importance.To understand the significance of differential equations (微分方程), one must first grasp the concept of derivatives. A derivative represents the rate of change of a function concerning its variables. For instance, in physics, the derivative of position with respect to time gives us velocity, while the derivative of velocity with respect to time gives us acceleration. Thus, differential equations (微分方程) allow us to describe systems where change is continuous and can be quantified mathematically.One of the simplest forms of a differential equation (微分方程) is the first-order linear differential equation, which can be expressed as dy/dx + P(x)y = Q(x). Here, P(x) and Q(x) are functions of x, and y is the unknown function we aim to solve for. This form of the equation can model various real-life situations, such as population growth or decay, where the rate of change of the population is proportional to its current size.In contrast, second-order differential equations (微分方程) involve second derivatives and are often used to describe more complex systems. For example, Newton's second law of motion can be represented using a second-order differential equation (微分方程), which relates force, mass, and acceleration. By solving these equations, engineers can design safer structures, predict the motion of objects, and even analyze electrical circuits.The process of solving differential equations (微分方程) can be quite intricate, depending on the equation's type and order. Some equations can be solved analytically, yielding exact solutions, while others may require numerical methods for approximation. Techniques such as separation of variables, integrating factors, and characteristic equations are commonly employed to tackle these mathematical challenges.Moreover, differential equations (微分方程) can also be classified into ordinary and partial differential equations. Ordinary differential equations (微分方程) involve functions of a single variable, while partial differential equations (微分方程) deal with functions of multiple variables. The latter is particularly significant in fields like fluid dynamics, where the behavior of fluid flow is described by the Navier-Stokes equations, a set of partial differential equations (微分方程).In conclusion, differential equations (微分方程) are an essential part of modern mathematics, providing the framework for understanding and predicting the behavior of dynamic systems. Their applications span across various disciplines, making them invaluable for scientists, engineers, and economists alike. As we continue to explore the complexities of the universe, mastering differential equations (微分方程) will remain a critical skill for anyone engaged in scientific inquiry or technological advancement.

微分方程是数学和工程学中的一个基本概念，是建模自然界各种现象的重要工具。differential equation（微分方程）是一个将函数与其导数联系起来的方程。这些方程在物理学、生物学、经济学甚至社会科学等众多领域中出现，展示了它们的多功能性和重要性。要理解differential equations（微分方程）的意义，首先必须掌握导数的概念。导数表示函数相对于其变量的变化率。例如，在物理学中，位置关于时间的导数给我们速度，而速度关于时间的导数则给我们加速度。因此，differential equations（微分方程）使我们能够描述变化是连续的并且可以用数学量化的系统。最简单的differential equation（微分方程）形式之一是第一阶线性微分方程，可以表示为dy/dx + P(x)y = Q(x)。在这里，P(x)和Q(x)是x的函数，而y是我们旨在求解的未知函数。该方程可以模拟多种现实生活中的情况，例如人口增长或衰退，其中人口的变化率与其当前规模成正比。相比之下，二阶differential equations（微分方程）涉及二阶导数，通常用于描述更复杂的系统。例如，牛顿第二运动定律可以用一个二阶differential equation（微分方程）来表示，该方程将力、质量和加速度联系起来。通过求解这些方程，工程师可以设计更安全的结构，预测物体的运动，甚至分析电路。解决differential equations（微分方程）的过程可能相当复杂，这取决于方程的类型和阶数。一些方程可以解析求解，得出精确解，而其他方程可能需要数值方法进行近似。分离变量法、积分因子法和特征方程法等技术通常被用来应对这些数学挑战。此外，differential equations（微分方程）还可以分为常微分方程和偏微分方程。常differential equations（微分方程）涉及单一变量的函数，而偏differential equations（微分方程）处理多个变量的函数。后者在流体动力学等领域尤为重要，流体流动的行为由Navier-Stokes方程组描述，这是一组偏differential equations（微分方程）。总之，differential equations（微分方程）是现代数学的重要组成部分，为理解和预测动态系统的行为提供了框架。它们的应用跨越多个学科，使其对科学家、工程师和经济学家而言都是不可或缺的。随着我们继续探索宇宙的复杂性，掌握differential equations（微分方程）将仍然是任何从事科学研究或技术进步的人的关键技能。