The author considers the solutions of an \(m\)th-order differential equation \[ w^{(m)}= Q(r,w, \dots, w^{(m-1)}), \] satisfying the condition: \[ w^{(m-i)}(r_*)> {t_*\over(i-1)!} r_*^{i-1}, \quad \text{for }i=1,2, \dots, m. \] If \(Q\) is a Carathéodory-type function on \([r_*,+\infty [\times \mathbb{R}^m\), such that for some \(k\in\{0,1, \dots, m-2\}\) the inequality \(Q(r,t_0, \dots, t_{m-1})\geq p(r)g(t_k)\) is satisfied on the set \(\{(r,t_0, \dots, t_{m-1})| r\geq r_*\) and \(t_{m-i}\geq{t_*\over(i-1)!} r^{i-1}\), for \(i=1,2, \dots, m\}\), then the following problem is investigated: under what assumptions on \(p\) and \(g\) each nonextendable solution is singular of second type? Some estimates for the \(k\)th-order derivative of a solution are given. No proofs are included.