In this paper, the quaternion differential equation, \({\textbf{A}}{\textbf{x}}'(t)+ {\textbf{B}}{\textbf{x}}(t)={\textbf{f}}(t)\) , is studied in the case when the quaternion matrix \(\textbf{A}\) is singular and \({\textbf{f}}(t)\) is a quaternion-valued vector function of a real variable t. Its solvability conditions are derived, and the general solution is represented by an explicit formula using generalized inverses of coefficient matrices. The dual (left) quaternion differential equation is explored as well. Also, we consider their partial cases when there is a constant quaternion matrix instead of a quaternion-valued vector function. At that, their general solutions are represented by closed formulas using the Moore-Penrose and Drazin inverses. To illustrate the effectiveness of these results, algorithms, and examples for finding solutions to considered differential quaternion matrix equations are provided. To compute generalized inverse matrices, we use a direct method, namely, their determinantal representations based on the theory of row-column noncommutative determinants previously introduced by the author.
