论二阶齐线性方程的两个不变量

The Two Invariants of Homogeneous Linear Differential Equations of Order Two

讨论了形如xn+P(t)x'+q(t)x=0的二阶齐线性方程,推广了文[1]中该方程两个不变量I(t)(?)-q(t)-1/4q2(t)-1/2p'(t)和J(t)(?)q'(t)+2p(t)q(t)/2|q(t)/3/2的证明,并由此推出几类齐线性方程可化为常系数方程或易求解的方程的情形及它的通解,推广了文[2] 的4种情形。

The paper discusses homogeneous linear differential equations of order two about xn+P(t)x'+q (t)x=0. It extends the proof of the two invariants in paper [2] which are I(t) (?)q(t)-1/42(t)-1/2p'(t) and J(t) (?) 2|q(t)|3/2/q'(t)+2q(t)q(t), and pullouts some homogeneous linear differential equations, which can be changed into constant coefficient equations or others solved easily, and ordinary solutions. Furthermore, this paper provides four conditions based on the conclusion in [2].