V2EX method of variation of parameters

Method of Variation of Parameters

Definition|释义

参数变易法(也称“常数变易法”):一种用于求解非齐次线性常微分方程(以及某些线性方程组)特解的方法。思路是把齐次方程解中原本的“常数”改为随自变量变化的函数,从而构造非齐次方程的解。(在不同教材中也可能强调其等价名称与推导形式。)

Pronunciation (IPA)|发音

/mθd v vrien v prmtrz/

Examples|例句

We solved the nonhomogeneous equation using the method of variation of parameters.
我们用参数变易法求解了这个非齐次方程。

In a second-order linear ODE, the method of variation of parameters expresses a particular solution in terms of the fundamental solutions and the Wronskian.
在二阶线性常微分方程中,参数变易法会用基本解与朗斯基行列式来表示一个特解。

Etymology|词源

该术语来自微分方程理论中的经典构造:在齐次线性方程的通解里,解通常写成若干基本解的线性组合,系数是常数;variation of parameters 字面意思是“让参数发生变化”,即把这些常数系数“变成函数”,再代回原方程求出这些函数,从而得到非齐次方程的特解。早期文献与教材中也常用 variation of constants(常数变易)来指同一思想。

Related Words|相关词

In Literature|文学与著作中的用例

  • Elementary Differential Equations and Boundary Value Problems(Boyce & DiPrima)常见章节专门介绍 variation of parameters/variation of constants。
  • Differential Equations with Applications and Historical Notes(George F. Simmons)以较强的历史脉络讲解该方法。
  • Advanced Engineering Mathematics(Erwin Kreyszig)工程数学语境下系统讲解并配套例题。
  • An Introduction to Ordinary Differential Equations(Coddington)更偏理论化地呈现线性理论与该构造方法。
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