constant coefficient

简明释义

常系数

英英释义

例句

1.The behavior of the system changes dramatically if we introduce a variable instead of a constant coefficient 常数系数.

如果我们引入一个变量而不是<常数系数>，系统的行为会发生显著变化。

2.The differential equation we are studying has a constant coefficient 常数系数, which simplifies the solution process.

我们正在研究的微分方程具有<常数系数>，这简化了解决过程。

3.In control theory, having a constant coefficient 常数系数 in the transfer function makes the analysis straightforward.

在控制理论中，传递函数中具有<常数系数>使得分析变得简单明了。

4.In linear algebra, a system of equations is easier to solve when it has a constant coefficient 常数系数.

在线性代数中，当方程组具有<常数系数>时，更容易求解。

5.When modeling physical systems, using a constant coefficient 常数系数 can lead to more accurate predictions.

在建模物理系统时，使用<常数系数>可以导致更准确的预测。

作文

In mathematics, particularly in the study of differential equations, the term constant coefficient refers to a situation where the coefficients of the terms in an equation do not change with respect to the variable. This concept is crucial because it simplifies the process of finding solutions to these equations. When we consider a linear differential equation with constant coefficient, we can apply various methods, such as the characteristic equation method, to solve for the unknown functions. For instance, consider the second-order linear differential equation given by: y'' + ay' + by = 0,where a and b are constants. Here, the coefficients of y'' and y' do not vary; they remain fixed regardless of the value of y. This allows us to use the characteristic equation, which is derived from assuming a solution of the form y = e^(rt), where r is a constant. By substituting this assumption into the differential equation, we can transform it into an algebraic equation, making it easier to solve.The importance of constant coefficient equations extends beyond pure mathematics; they are also prevalent in physics and engineering. For example, in mechanical systems, the motion of a mass-spring-damper system can often be modeled using differential equations with constant coefficient. The damping ratio and natural frequency of the system can be determined from the coefficients of the equation, allowing engineers to design systems that behave predictably under various conditions.Moreover, the solutions to these equations can reveal critical information about the stability and behavior of the system over time. If the roots of the characteristic equation are real and distinct, the system will exhibit different types of damping. If the roots are complex, the system will oscillate, which is a key factor in understanding dynamic systems.In contrast, when dealing with equations that have variable coefficients, the complexity increases significantly. Solutions may require special functions or numerical methods, and the behavior of the system can become unpredictable. Therefore, the study of constant coefficient differential equations serves as a foundation for more advanced topics in applied mathematics.In conclusion, understanding the concept of constant coefficient is essential for anyone delving into the realms of mathematics, physics, or engineering. It not only simplifies the problem-solving process but also provides insights into the behavior of systems described by these equations. As we encounter increasingly complex scenarios in various fields, the principles grounded in constant coefficient equations will continue to play a vital role in our understanding and application of mathematical concepts. In summary, a constant coefficient is a fixed value that does not change with the variable in the context of equations, allowing for more straightforward solutions and analyses in various scientific and engineering applications.

在数学中，特别是在微分方程的研究中，术语常数系数指的是方程中各项的系数不随变量变化的情况。这个概念至关重要，因为它简化了求解这些方程的过程。当我们考虑一个具有常数系数的线性微分方程时，我们可以应用各种方法，例如特征方程法，来求解未知函数。例如，考虑以下二阶线性微分方程：y'' + ay' + by = 0,其中a和b是常数。在这里，y''和y'的系数保持不变；它们无论y的值如何都保持固定。这使得我们可以使用特征方程，该方程是通过假设y的形式为y = e^(rt)而得出的，其中r是一个常数。通过将这一假设代入微分方程，我们可以将其转化为代数方程，从而简化求解过程。常数系数方程的重要性超越了纯数学的范畴；它们在物理学和工程学中也很常见。例如，在机械系统中，质量-弹簧-阻尼器系统的运动通常可以用具有常数系数的微分方程进行建模。系统的阻尼比和自然频率可以从方程的系数中确定，使工程师能够设计在各种条件下表现出可预测行为的系统。此外，这些方程的解可以揭示系统随时间变化的稳定性和行为的重要信息。如果特征方程的根是实数且不同，系统将表现出不同类型的阻尼。如果根是复数，系统将振荡，这是理解动态系统的关键因素。相比之下，当处理具有可变系数的方程时，复杂性显著增加。解可能需要特殊函数或数值方法，系统的行为可能变得不可预测。因此，研究常数系数微分方程为更高级的话题奠定了基础。总之，理解常数系数的概念对于任何深入数学、物理或工程领域的人来说都是必不可少的。它不仅简化了问题解决过程，还提供了对这些方程描述的系统行为的洞察。随着我们在各个领域遇到越来越复杂的场景，基于常数系数方程的原理将继续在我们对数学概念的理解和应用中发挥重要作用。总之，常数系数是指在方程上下文中不随变量变化的固定值，这使得在各种科学和工程应用中更容易得到解决方案和分析。