This work is concerned with the ultimate boundedness of solutions of the system of vector differential equations \begin{equation*} \dot{X}=H(Y),\quad \dot{Y}=-F(X,Y)Y-G(X)+P(t,X,Y), \end{equation*} where $t\in\mathbb{R}^+,\ X=X(t)$, $Y=Y(t)\in \mathbb{R}^n$, $F:\mathbb{R}^n×\mathbb{R}^n\to\mathbb{R}^{n×n}, \ G,H:\mathbb{R}^n\to\mathbb{R}^n \ \mbox{and} \ P:\mathbb{R}^+×\mathbb{R}^n×\mathbb{R}^n\to\mathbb{R}^n.$ By using a Lyapunov function as a basic technique, we prove that the solutions of the system of equations are ultimately bounded. In addition, result obtained includes and improves some related results in literature.