从齐次方程的一个特解到非齐次方程的通解——二阶线性微分方程的又一种常数变易法

From One Special Solution of Homogeneous Linear Differential Equation to the General Solution of Non-homogeneous Linear Differential Equation

介绍了二阶线性微分方程的又一种常数变异法。其基本思想就是:首先,通过观察法可以得到满足某些特定条件的线性齐次微分方程的一个特解;其次,通过这一个特解用降阶法得到另外一个与之线性无关的特解,从而得到线性齐次方程的通解;最后,通过一种常数变异法得到对应非齐次方程的通解。

In this paper, we introduced another method of variation of constant of the second-order linear equation. Its basic ideas are as follows： firstly, by observing, we get a special solution of linear homogeneous differential equation satisfying some conditions; secondly, we also get another special linearly independent solution by the method of declining the rank, then the general solutions of this differential equation are derived; finally, the general solutions of the corresponding non-homogeneous differential equation are also derived.