On the generalized Apostol-Kolodner differential equation of the second-order

This paper concerns another approach for solving the matrix differential equations of the second order $X''(t)=AX'(t)+BX(t)$, where $A$, $B$ are square matrices of order ${d\times d}$. Such approach is based on some properties of matrix square root and the linear difference equation. We establish various new results and explicit formulas for the solutions of this type of matrix differential equations. Moreover, because of the non-commutativity condition between $A$ and $B$, the matrix fundamental system will play an important role. Finally, our results and their robustness, are validated by providing some examples and applications.