V2EX variation of parameters

Variation of Parameters

Definition / 定义

Variation of parameters(参数变易法):常微分方程中的一种求解方法,主要用于求非齐次线性微分方程的特解。核心思想是:把齐次方程解中的“常数”改为随自变量变化的函数,从而构造非齐次方程的解。(也常见于二阶及更高阶线性方程、以及线性方程组。)

Pronunciation / 发音(IPA)

/vrien v prmtrz/

Examples / 例句

We used variation of parameters to find a particular solution.
我们用参数变易法找到了一个特解。

After finding the complementary solution, she applied variation of parameters to solve the nonhomogeneous second-order equation with a trigonometric forcing term.
在求出互补解之后,她用参数变易法来求解带有三角函数外力项的非齐次二阶方程。

Etymology / 词源

variation 源自拉丁语 variatio,含义是“变化、变动”;parameter 来自希腊语词根 *para-*(旁边、辅助)与 metron(度量),合起来有“用来度量/描述的量”的意思。作为数学方法,“数变易法”常与拉格朗日(Lagrange)联系在一起:他在研究线性微分方程时系统发展了这种“把常数变成函数”的构造思路。

Related Words / 相关词汇

Notable Works / 文献与作品中的使用

  • Elementary Differential Equations and Boundary Value Problems(Boyce & DiPrima)
  • A First Course in Differential Equations(Dennis G. Zill)
  • Differential Equations with Applications and Historical Notes(George F. Simmons)
  • Ordinary Differential Equations(E. L. Ince)
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