On the Oscillation of Certain Third Order Linear Differential Equations

Abstract

This paper is concerned with the relationship between the concepts of oscillation, nonoscillation and disconjugacy of the general third order linear differential equation $y''’ + p(x)y'' + q(x)y’ + r(x)y = 0$. Two cases are considered: (i) $q(x)$ nonpositive and $r(x)$ nonnegative and (ii) both $q(x)$ and $r(x)$ nonpositive. Sufficient conditions for the disconjugacy of the equation are presented, improving the recent work of W. J. Kim. The relationship between disconjugacy and oscillation is described in each case, and these results provide a partial answer to a question raised by J. H. Barrett.