particular solution

简明释义

特解

英英释义

例句

1.In solving the differential equation, we found a particular solution that fits the initial conditions.

在求解微分方程时，我们找到了一个符合初始条件的特解。

2.In physics, finding a particular solution can help us understand specific scenarios like projectile motion.

在物理学中，找到一个特解可以帮助我们理解诸如抛体运动等具体场景。

3.The mathematician demonstrated how to derive a particular solution from the general form of the equation.

数学家展示了如何从方程的一般形式推导出一个特解。

4.The engineer proposed a particular solution for the stress analysis of the beam under load.

工程师提出了一个针对梁在载荷下应力分析的特解。

5.After several iterations, the team reached a particular solution to optimize the production process.

经过几次迭代，团队达成了一个优化生产过程的特解。

作文

In mathematics, particularly in the field of differential equations, the term particular solution refers to a specific solution that satisfies both the differential equation and any initial or boundary conditions imposed on the problem. This is in contrast to the general solution, which encompasses all possible solutions of the differential equation. Understanding the concept of particular solution is crucial for solving real-world problems where specific conditions must be met.For instance, consider a simple first-order linear differential equation that models the cooling of an object. The general solution might describe how the temperature of the object changes over time under various conditions. However, if we have specific information, such as the initial temperature of the object and the ambient temperature of the environment, we can derive a particular solution that accurately predicts the object's temperature at any given time.The process of finding a particular solution often involves using methods such as undetermined coefficients or variation of parameters. These techniques allow mathematicians and engineers to tailor the general solution to fit the unique circumstances of a given scenario. The importance of obtaining a particular solution cannot be overstated, as it provides the precise information needed to make informed decisions based on mathematical models.Moreover, the relevance of particular solution extends beyond pure mathematics into fields such as physics, engineering, and economics. For example, in physics, the motion of a pendulum can be described by a differential equation. While the general solution may provide insight into the pendulum's behavior, the particular solution will reveal the exact path of the pendulum when initial conditions, such as its starting angle and velocity, are known.In engineering, designing structures requires understanding how materials will respond to various forces. By applying the principles of differential equations, engineers can derive particular solutions that predict stress and strain in materials under specific loads and conditions. This ensures safety and efficiency in construction and manufacturing processes.Economists also rely on particular solutions when modeling economic behaviors and trends. For instance, when analyzing market responses to changes in policy or external factors, economists use differential equations to represent these dynamics. The particular solution derived from their model can help predict how a specific market will react, allowing policymakers to make data-driven decisions.In conclusion, the concept of particular solution plays a vital role in various scientific and engineering disciplines. It allows for the application of mathematical theories to real-world problems, providing precise answers that inform decision-making. Whether in predicting the cooling of an object, the motion of a pendulum, or the response of an economy, the ability to find a particular solution is essential for achieving accurate and reliable results. As we continue to explore complex systems in nature and society, mastering the use of particular solutions will remain a fundamental skill for students and professionals alike.

在数学中，特别是在微分方程领域，术语特解指的是满足微分方程及其施加的任何初始或边界条件的特定解。这与一般解形成对比，一般解包含了微分方程的所有可能解。理解特解的概念对于解决必须满足特定条件的现实问题至关重要。例如，考虑一个简单的一阶线性微分方程，它模拟一个物体的冷却过程。一般解可能描述物体温度随时间变化的情况，但如果我们有特定的信息，例如物体的初始温度和环境的温度，我们可以推导出一个特解，准确预测物体在任何给定时间的温度。寻找特解的过程通常涉及使用未定系数法或参数变换法等方法。这些技术使数学家和工程师能够根据给定场景的独特情况调整一般解。获得特解的重要性不言而喻，因为它提供了基于数学模型做出明智决策所需的精确信息。此外，特解的相关性不仅限于纯数学，还扩展到物理、工程和经济学等领域。例如，在物理学中，摆的运动可以用微分方程来描述。虽然一般解可能提供关于摆行为的见解，但特解将揭示当已知初始条件（如起始角度和速度）时，摆的确切路径。在工程中，设计结构需要理解材料如何响应各种力量。通过应用微分方程的原理，工程师可以推导出特解，预测在特定负载和条件下材料的应力和应变。这确保了建筑和制造过程的安全性和效率。经济学家在建模经济行为和趋势时也依赖于特解。例如，在分析市场对政策或外部因素变化的反应时，经济学家使用微分方程来表示这些动态。从他们的模型中推导出的特解可以帮助预测特定市场的反应，使政策制定者能够做出基于数据的决策。总之，特解的概念在各个科学和工程学科中发挥着至关重要的作用。它使数学理论能够应用于现实问题，提供精确的答案以指导决策。无论是预测物体的冷却、摆的运动，还是经济的反应，找到特解的能力对实现准确可靠的结果至关重要。随着我们继续探索自然和社会中的复杂系统，掌握使用特解的技能将始终是学生和专业人士的重要能力。