method of undeterminated coefficients

简明释义

待定系数法

英英释义

例句

1.The professor demonstrated the method of undetermined coefficients 未定系数法 during the lecture on differential equations.

教授在微分方程的讲座中演示了未定系数法。

2.Using the method of undetermined coefficients 未定系数法, we can easily solve for the constants in the equation.

使用未定系数法，我们可以轻松求解方程中的常数。

3.In our project, we relied on the method of undetermined coefficients 未定系数法 to model the system's behavior.

在我们的项目中，我们依赖于未定系数法来建模系统的行为。

4.In solving linear differential equations, the engineer applied the method of undetermined coefficients 未定系数法 to find a particular solution.

在求解线性微分方程时，工程师应用了未定系数法来找到特解。

5.The method of undetermined coefficients 未定系数法 is particularly useful when the non-homogeneous part is a polynomial.

当非齐次部分是多项式时，未定系数法特别有用。

作文

The study of differential equations is a fundamental aspect of mathematics that has applications across various fields, including physics, engineering, and economics. Among the many techniques used to solve these equations, the method of undetermined coefficients stands out as a particularly useful approach for solving linear ordinary differential equations with constant coefficients. This method allows us to find particular solutions to non-homogeneous equations by assuming a specific form for the solution and determining the coefficients through substitution and simplification.To understand the method of undetermined coefficients, we first need to recognize the structure of the differential equation we are dealing with. A typical linear non-homogeneous differential equation can be expressed in the form: Ly = f(x), where L is a linear differential operator, y is the unknown function, and f(x) is a known function that represents the non-homogeneous part. The general strategy involves two main steps: first, we find the complementary solution (also known as the homogeneous solution) by solving the associated homogeneous equation Ly = 0. Then, we apply the method of undetermined coefficients to find a particular solution to the non-homogeneous equation.The beauty of the method of undetermined coefficients lies in its simplicity. We start by guessing a form for the particular solution based on the type of function f(x). For instance, if f(x) is a polynomial, we assume a polynomial solution of the same degree. If f(x) is an exponential function, we assume a solution of the same exponential form. Similarly, for trigonometric functions, we assume a combination of sine and cosine functions. The key is to include undetermined coefficients in our guess, which we will later solve for.Once we have our assumed form for the particular solution, we substitute it back into the original differential equation. This substitution will yield an equation that relates the coefficients of our assumed solution to the coefficients of the known function f(x). By equating the coefficients of like terms, we can solve for the undetermined coefficients. This process often involves algebraic manipulation, but it is generally straightforward.After determining the coefficients, we can then write down the complete solution to the differential equation by combining the complementary solution and the particular solution obtained from the method of undetermined coefficients. This complete solution provides us with a comprehensive understanding of the behavior of the system described by the differential equation.In conclusion, the method of undetermined coefficients is a powerful technique for solving linear non-homogeneous differential equations. Its effectiveness lies in its straightforward application and the ability to derive solutions systematically. Understanding this method not only enhances our mathematical toolkit but also deepens our appreciation for the interconnectedness of different mathematical concepts. As we continue to explore the vast landscape of mathematics, the method of undetermined coefficients remains a valuable resource for tackling complex problems and advancing our knowledge in this essential field.

微分方程的研究是数学的一个基本方面，它在物理、工程和经济学等多个领域都有应用。在解决这些方程的众多技术中，未定系数法作为一种特别有用的方法，突显出其在求解具有常系数的线性常微分方程方面的重要性。该方法通过假设特定形式的解来寻找非齐次方程的特解，并通过代入和简化来确定系数。要理解未定系数法，我们首先需要认识到我们所处理的微分方程的结构。一个典型的线性非齐次微分方程可以表示为：Ly = f(x)，其中L是线性微分算子，y是未知函数，f(x)是已知函数，表示非齐次部分。一般策略包括两个主要步骤：首先，通过求解相关的齐次方程Ly = 0来找到补充解（也称为齐次解）。然后，我们应用未定系数法来寻找非齐次方程的特解。未定系数法的美在于其简单性。我们首先根据f(x)的类型猜测特解的形式。例如，如果f(x)是多项式，我们假设一个相同次数的多项式解。如果f(x)是指数函数，我们假设一个相同指数形式的解。类似地，对于三角函数，我们假设正弦和余弦函数的组合。关键是要在我们的猜测中包含未定系数，我们将在后面为其求解。一旦我们得到了假设的特解形式，就将其代入原微分方程。这种代入将产生一个将我们假设的解的系数与已知函数f(x)的系数联系起来的方程。通过对同类项的系数进行比较，我们可以解出未定系数。这个过程通常涉及代数运算，但一般来说是直接的。确定系数后，我们可以通过将补充解和通过未定系数法获得的特解结合起来，从而写出微分方程的完整解。这个完整解为我们提供了对描述微分方程所描述的系统行为的全面理解。总之，未定系数法是一种强大的技术，用于解决线性非齐次微分方程。它的有效性在于其简单的应用和系统地推导解的能力。理解这一方法不仅增强了我们的数学工具箱，还加深了我们对不同数学概念之间相互联系的欣赏。随着我们继续探索数学的广阔领域，未定系数法仍然是解决复杂问题和推进我们在这一重要领域知识的宝贵资源。