On the instability of trivial solutions of a class of eighth-order differential equations (Q810719)

Consider the eighth-order differential equation \[ (1)\quad x^{(8)}+\sum^{8}_{i=1}a_ iD^{(8-i)}x=0\quad (D\equiv d/dt) \] with constant coefficients \(a_ i\) \((i=1,2,...,8)\). It is shown that, in the special case \(a_ 1=0\) and \(a_ 8\neq 0\), the trivial solution of (1) is unstable if \(a_ 3\neq 0\) and \((a_ 7-(1/4)a_ 3^{-1}a^ 2_ 5)sgn a_ 3>0\), but the main objective of the paper is to generalize this result to an equation (1) in which \(a_ 5\), \(a_ 6\), \(a_ 7\) and \(a_ 8\) are not necessarily constants. The proof for the nonlinear case is based on a well known instability criterion of Krasovskij.