Riccati方法与微分方程的渐近积分(英文)

Riccati Techniques and Asymptotic Integrations of Differential Equations

考虑二阶常微分方程x″+f(t)x=0,t≥a,(1)假设应用Riccati方法得到方程(1)的主解(principal solution)的一个渐近积分并研究其副解(nonprincipal solutions)的三种不同的渐近性质.主要结果如下:定理1 若(Ⅰ)成立,则方程(1)有解x_1满足及另一解x_2满足x_2(t)=t[1+o(1)]. 反之,若方程(1)有解x(t)→1,t→∞,则(Ⅰ)成立. 定理2 设(Ⅰ)成立.(i)若(Ⅱ)成立,则方程(1)有解x_2使x_2’(t)=1+[tF(t)+G(t)][1+o(1)]+o(1). (ii) 反之,若方程(1)有解x使x’→1,t→∞,则(Ⅱ)成立. 定理3 若(Ⅲ)和(Ⅳ)成立,则方程(1)有解x_1满足(2)

Considering the second order differential equationx'+f(t)x=0, where f(t) is twice integrable (perhaps conditionally) at t=∞, the author employs Riccati techniques to obtain an asymptotic integration of a principal solution x_1(t) of (1) and studies the asymptotic behaviour of nonprincipal solutions.