二阶非齐次微分方程解的平方可积性与有界性

Quadratic Integrability and Boundedness of Solutions of a Class of Linear Nonhomogeneous Differential Equations of Second Order

本文借助于一类带有参数 m,n∈ R的辅助函数，得到了二阶非齐次线性微分方程 (r(t)x′ )′＋ p(t)x′＋ [q1(t)＋ q2(t)]x=f(t) 的所有解均平方可积及所有解都有界的判定准则。所得结果改进和推广了现有的许多判定准则。

Consider the nonhomogeneous linear differential equation (r(t)x′ )′＋ p(t)x′＋ [q1(t)＋ q2(t)]x=f(t)(1) By virtue of an auxiliary function with parameters m,n∈ R,we obtain some sufficient conditions under which all solutions of equation (1) belong to L∞ [a,∞ )∩ L2[a,∞ ).Our results improve and generalize some existing theorems.