The author considers the differential equation \((1)\quad (q(qy')')'+(1/2)((q_ 1y)'+q_ 1y')+(p_ 0y')'+p_ 1y=0,\) where \(q,q_ 1,p_ 0,p_ 1\) are defined on [a,\(\infty)\) and are not necessarily real-valued, while q is nowhere zero in this interval. Let \(\lambda_ i=\lambda_ i(t)\) \((i=1,2,3)\) be roots of the equation \(q^ 2\lambda^ 3+p_ 0\lambda^ 2+q_ 1\lambda +p_ 1=0.\) There are given sufficient conditions which ensure that equation (1) has three linearly independent solutions \[ y_ k(t)=(q(t)p_ 1(t))^{(- 1/3)}\quad (1+o(1))\exp (\int^{t}_{a}\lambda_ k(s)ds),\quad k=1,2,3. \] The results are extended to higher odd-order differential equations.