Results for Sixth Order Positively Homogeneous Equations

Abstract

We consider positively homogeneous the sixth order differential equations of the type x (6) = h(t, x), where h possesses the property that h(t, cx) = ch(t, x) for c ≥ 0. This class includes the linear equations x (6) = p(t)x and piece-wise linear ones x (6) = k2 x + − k1 x − . We consider conjugate points and angles associated with extremal solutions and prove some comparison results.Key words: differential equations of 6-th order, equations with asymmetric nonlinearities, conjugate points, positive homogeneous equations.

Periodic solutions for a class of perturbed sixth-order autonomous differential equations

“…In general, obtaining analytically periodic solutions of a differential system is a very difficult task, usually impossible. Recently, the study of the periodic solutions of sixth-order of DEs has been considered by several authors (see Refs [3,20,21]). Here, using the averaging theory, we reduce this difficult problem for the differential equation (3) to find the zeros of a nonlinear system of five equations.…”