We propose efficient algorithms for solving large-scale matrix differential equations. In particular, we deal with Riccati differential equations (RDEs) and Lyapunov differential equations (LDEs). We focus on methods, based on standard versions of ordinary differential equations, in the matrix setting. The application of these methods yields algebraic Lyapunov equations (ALEs) with a certain structure to be solved in every step. The alternating direction implicit (ADI) algorithm and Krylov subspace based methods allow to exploit this special structure. However, a direct application of classic low-rank formulations requires the use of complex arithmetic. Using an LDLT -type decomposition of both, the right hand side and the solution of the equation we avoid this problem. Thus, the proposed methods are a more practical alternative for large-scale problems arising in applications. Also, they make feasible the application of higher order methods. The numerical results show the better performance of the proposed methods compared to earlier formulations.