Instability of solution for the third order linear differential equation with varied coefficient (Q1177876)

The paper is concerned with the equation (1) \(\dddot x+a(t)\ddot x+b(t)\dot x+c(t)x=0\). Here the dots denote differentiation with respect to \(t\), and \(a\), \(b\) and \(c\) depend only on \(t\) and are differentiable and bounded and satisfy \(a(t)b(t)-c(t)\neq 0\) for all \(t\geq t_ 0\) (\(t_ 0\) sufficiently large). The object is to show that if at least one of the roots \(\lambda_ 1\), \(\lambda_ 2\), \(\lambda_ 3\) of the equation \(\lambda^ 3+a(t)\lambda^ 2+b(t)\lambda+c(t)=0\) has a positive real part, then the trivial solution of (1) is unstable provided that \(\max_{t\geq t_ 0}(|\dot a|,|\dot b|,|\dot c|)\) is sufficiently small. The proof involves the use of Lyapunov's second method applied to the three cases: (i) each of \(\lambda_ 1\), \(\lambda_ 2\) and \(\lambda_ 3\) has a positive real part, (ii) two \(\lambda_ 1\), \(\lambda_ 2\) and \(\lambda_ 3\) have positive real parts, (iii) one (only) of \(\lambda_ 1\), \(\lambda_ 2\) and \(\lambda_ 3\) has a positive real part. The proof for the cases (ii) and (iii) makes use of the additional restriction \(a(t)+c(t)+b(t)c(t)<0\) or \(>0\).