一阶、二阶常微分方程简捷解法

Solving Process Simply for First-Order and Second-Order Ordinary Differential Equation

本文针对一阶、二阶可降阶、二阶常系数线性常微分方程的常规解法进行梳理,对题型加以细化,并针对每一种细化题型总结出一套较为独特简捷的解法。创新之处在于针对一阶线性微分方程三种题型直接凑微分,二阶可降阶微分方程不设中间变量直接凑微分,二阶常系数线性常微分方程三种题型(特征方程单根、二重根、共轭复根)直接凑微分求通解,二阶常系数非齐次线性微分方程三种基本题型及四种扩展题型直接求特解,解题方法快速简洁,深受学生好评。

This paper combs the conventional solving process for first-order, second-order reducible, se-cond-order constant coefficient linear ordinary differential equation, refines the types of problems, and summarizes a set of unique and simple solving process for each refined type of problems. The innovation lies in the direct integration of three types of problems of the first order linear differential equation, the second order reducible differential equation without intermediate vari-ables, the second order constant coefficient linear ordinary differential equation with three types of problems (single root, double root and conjugate complex root of characteristic equation), the second order constant coefficient linear non-homogeneous differential equation with three basic problems and four extended problems, and the direct integration of differential equations to find general solutions. The solution method is fast and simple, and praised by students highly.