differential equation

简明释义

微分方程式

英英释义

例句

1.A simple harmonic oscillator can be described by a second-order differential equation 微分方程.

简单谐振子可以用一个二阶differential equation 微分方程 来描述。

2.Engineers use differential equations 微分方程 to predict the behavior of systems under various conditions.

工程师使用differential equations 微分方程 来预测系统在不同条件下的行为。

3.The spread of diseases can often be modeled using a differential equation 微分方程 that accounts for the rate of infection.

疾病的传播通常可以用一个differential equation 微分方程 来建模，该方程考虑了感染的速率。

4.In physics, the motion of a falling object can be described by a differential equation 微分方程 that relates its velocity to time.

在物理学中，物体下落的运动可以通过一个differential equation 微分方程 来描述，该方程将其速度与时间联系起来。

5.To solve a differential equation 微分方程, one must often apply initial conditions to find a unique solution.

要解决一个differential equation 微分方程，通常需要应用初始条件以找到唯一解。

作文

Differential equations are a fundamental aspect of mathematics and engineering, providing a framework for modeling various phenomena in the real world. A differential equation (微分方程) is an equation that relates a function with its derivatives. In simpler terms, it describes how a quantity changes over time or space, which is crucial for understanding dynamic systems. For instance, in physics, the motion of an object can be described using differential equations (微分方程), where the position of the object is a function of time, and its velocity and acceleration are represented by the first and second derivatives of that function, respectively.One of the most famous examples of a differential equation (微分方程) is Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, this can be expressed as F = ma, where 'F' is the force, 'm' is the mass, and 'a' is the acceleration. If we express acceleration as the derivative of velocity with respect to time, we arrive at a differential equation (微分方程) that can be solved to find the object's velocity and position over time.The importance of differential equations (微分方程) extends far beyond physics; they are used in biology to model population dynamics, in economics to describe market behaviors, and in engineering to analyze systems and processes. For example, the logistic growth model, which describes how populations grow rapidly at first and then slow down as they reach their carrying capacity, is governed by a differential equation (微分方程). This model helps ecologists understand and predict changes in population sizes under different environmental conditions.Moreover, differential equations (微分方程) can be classified into several types, including ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve functions of a single variable and their derivatives, while PDEs involve multiple variables. Each type has its own methods of solution, ranging from analytical techniques to numerical simulations. For instance, the heat equation, which describes how heat diffuses through a medium, is a classic example of a partial differential equation (微分方程) that arises in thermodynamics.In modern applications, the use of differential equations (微分方程) has become increasingly sophisticated due to advancements in computational technology. Numerical methods allow scientists and engineers to solve complex differential equations (微分方程) that cannot be solved analytically, enabling them to model intricate systems such as weather patterns, fluid dynamics, and even financial markets. These numerical solutions provide valuable insights and predictions that inform decision-making across various fields.In conclusion, differential equations (微分方程) are not just abstract mathematical concepts; they are powerful tools that help us understand and predict the behavior of dynamic systems in the world around us. Their applications span numerous disciplines, making them essential for anyone seeking to comprehend the complexities of natural and artificial systems. As we continue to push the boundaries of knowledge in science and technology, the role of differential equations (微分方程) will undoubtedly remain pivotal in shaping our understanding of the universe.

微分方程是数学和工程学的一个基本方面，为建模现实世界中各种现象提供了框架。differential equation（微分方程）是一个将函数与其导数联系起来的方程。简单来说，它描述了某个量随时间或空间变化的方式，这对于理解动态系统至关重要。例如，在物理学中，物体的运动可以通过differential equations（微分方程）来描述，其中物体的位置是时间的函数，而其速度和加速度分别由该函数的一阶和二阶导数表示。一个著名的differential equation（微分方程）示例是牛顿第二运动定律，该定律指出作用在物体上的力等于物体的质量乘以其加速度。数学上可以表示为F = ma，其中'F'是力，'m'是质量，'a'是加速度。如果我们将加速度表示为速度关于时间的导数，我们就得到了一个可以解决的differential equation（微分方程），以找到物体随时间变化的速度和位置。differential equations（微分方程）的重要性远不止于物理学；它们在生物学中用于建模种群动态，在经济学中描述市场行为，在工程学中分析系统和过程。例如，逻辑斯蒂增长模型描述了种群如何在最初快速增长，然后在达到其承载能力时减缓，其由一个differential equation（微分方程）所支配。该模型帮助生态学家理解和预测在不同环境条件下种群数量的变化。此外，differential equations（微分方程）可以分为几种类型，包括常微分方程（ODE）和偏微分方程（PDE）。ODE涉及单变量及其导数的函数，而PDE涉及多个变量。每种类型都有自己的求解方法，从解析技术到数值模拟。例如，热方程描述了热量如何在介质中扩散，是热力学中经典的偏differential equation（微分方程）示例。在现代应用中，由于计算技术的进步，differential equations（微分方程）的使用变得越来越复杂。数值方法使科学家和工程师能够解决无法解析求解的复杂differential equations（微分方程），使他们能够建模天气模式、流体动力学甚至金融市场等复杂系统。这些数值解提供了有价值的见解和预测，指导各个领域的决策。总之，differential equations（微分方程）不仅仅是抽象的数学概念；它们是帮助我们理解和预测周围世界中动态系统行为的强大工具。它们的应用跨越众多学科，使其成为任何希望理解自然和人工系统复杂性的人必不可少的工具。随着我们继续推动科学和技术知识的边界，differential equations（微分方程）的角色无疑将在塑造我们对宇宙的理解中保持关键作用。