On the factorizations of ordinary linear differential operators

Abstract

ABSTRACT. Relations are found between the nonvanishing of certain Wronskians and disconjugacy properties of Lny + py = 0, where Lny is a disconjugate operator and p is sign definite. The results are then used to show ways in which Lny + py can be factored.Introduction.In this paper we investigate possible ways of factoring certain nth order linear differential operators of the form

Zeros of solutions and of Wronskians for the differential equation $L\sb ny+p(x)y=0$

“…Their investigation was based on relations between the nonvanishing of certain minors of the Wronskian and disconjugacy properties of equation (1.1). In this paper we extend the results of [4] and study, more generally, how many zeros certain minors of the Wronskian have. We shall show that certain odd order minors of the Wronskian and related solutions of (1.1) have the same number of zeros.…”

“…It is proved in [1,4] that the set {u o ' .,. , Un_I} which is defined in this way, exists, it is unique, and it is a basis of solutions of (1.1).…”

“…In a recent work [4], Etgen, Jones, and Taylor investigated ways of factoring the differential equation…”

“…Let the disconjugate differential operator Ln be Our results will be formulated in terms of a special basis {uo(x) , ... , Un_I (x)} of the solution space of (1.1), which has been used in [1][2][3][4][5]. Since uo(x) , ... , Un_I (x) will be defined in the next section by boundary value problems (2.2)0-(2.2)n_1 ' we first mention a known property [7] of boundary value problems of equation (1.1): If k quasiderivatives of a solution y vanish at the endpoint a and n -k quasiderivatives of y vanish at b, a < b , then n -k must be odd if p(x) > 0 and even if p(x) < O.…”

“…In [4], the authors studied the nonvanishing of certain minors of odd order of the Wronskian. Theorems 7, 8 and 13, 14 of [4] …”