variation of constants

简明释义

带数变值法

英英释义

例句

1.In solving differential equations, we often use the method of undetermined coefficients to find the variation of constants 常数变异.

在解决微分方程时，我们通常使用未定系数法来寻找常数变异。

2.Using variation of constants 常数变异, we can find particular solutions that fit initial conditions.

通过使用常数变异，我们可以找到适合初始条件的特解。

3.When applying the variation of constants 常数变异 method, we substitute the constants with functions of time.

在应用常数变异方法时，我们将常数替换为时间的函数。

4.The concept of variation of constants 常数变异 is crucial in understanding non-homogeneous differential equations.

理解非齐次微分方程时，常数变异的概念至关重要。

5.The variation of constants 常数变异 technique allows us to adjust parameters dynamically in our model.

使用常数变异技术使我们能够动态调整模型中的参数。

作文

In the realm of mathematics and physics, the concept of variation of constants plays a crucial role in understanding how systems behave under different conditions. The term refers to the method of solving differential equations where certain constants within the equation are allowed to vary, rather than remaining fixed. This approach can yield solutions that are more representative of real-world phenomena, especially in dynamic systems where parameters may change over time. For instance, consider the motion of a pendulum. The simple harmonic motion can be described by a differential equation involving constants such as mass and length. However, if we introduce factors like air resistance or varying gravitational force, the constants in our equation must adapt accordingly. Here, the variation of constants allows us to create a more accurate model that reflects these influences.Moreover, the variation of constants technique is not limited to physics; it extends into fields like economics and biology. In economics, for example, the demand and supply equations can include constants that represent consumer preferences or production costs. As these factors fluctuate due to market dynamics, applying the variation of constants helps economists predict changes in market equilibrium more effectively.In biology, the growth rates of populations can also be modeled using differential equations. If we consider a species that experiences changes in its environment, such as food availability or predation pressure, the constants that define its growth rate may need to vary. By employing the variation of constants, biologists can better understand the resilience of species and their ability to adapt to changing ecosystems.The importance of variation of constants is further emphasized when we explore its applications in engineering. Engineers often face complex systems where multiple variables interact. For example, in structural engineering, the load-bearing capacity of materials can change based on environmental factors like temperature and humidity. By applying the variation of constants approach, engineers can design structures that are not only safe but also efficient, taking into account the variability in material properties under different conditions.In summary, the variation of constants is a powerful tool in mathematical modeling that allows for a more nuanced understanding of complex systems across various disciplines. By recognizing that constants are not always constant, we can develop models that are more reflective of the realities we observe in nature and society. This flexibility is essential for advancing knowledge and solving practical problems, making the variation of constants an invaluable concept in both theoretical and applied sciences.

在数学和物理学领域，常数的变化这一概念在理解系统如何在不同条件下表现方面发挥着至关重要的作用。这个术语指的是解决微分方程的方法，其中方程中的某些常数被允许变化，而不是保持固定。这种方法可以产生更能代表现实世界现象的解，尤其是在动态系统中，参数可能随时间变化。例如，考虑一个摆的运动。简单谐运动可以通过涉及质量和长度的微分方程来描述。然而，如果我们引入空气阻力或变化的重力等因素，我们方程中的常数必须相应调整。在这里，常数的变化使我们能够创建一个更准确的模型，反映这些影响。此外，常数的变化技术不仅限于物理学；它还扩展到经济学和生物学等领域。例如，在经济学中，需求和供给方程可以包括表示消费者偏好或生产成本的常数。随着这些因素因市场动态而波动，应用常数的变化有助于经济学家更有效地预测市场均衡的变化。在生物学中，种群的生长率也可以通过微分方程建模。如果我们考虑一个物种在环境中经历变化，例如食物的可用性或捕食压力，那么定义其生长率的常数可能需要变化。通过采用常数的变化，生物学家可以更好地理解物种的韧性及其适应变化生态系统的能力。在工程学中，常数的变化的重要性进一步得到了强调。工程师经常面临复杂的系统，其中多个变量相互作用。例如，在结构工程中，材料的承载能力可能会根据温度和湿度等环境因素而变化。通过应用常数的变化方法，工程师可以设计出不仅安全而且高效的结构，考虑到材料在不同条件下的可变性。总之，常数的变化是数学建模中的一种强大工具，它允许对各个学科复杂系统进行更细致的理解。通过认识到常数并不总是恒定的，我们可以开发出更能反映我们在自然和社会中观察到的现实的模型。这种灵活性对于推动知识进步和解决实际问题至关重要，使得常数的变化成为理论和应用科学中不可或缺的概念。