  In the present paper, we propose Krylov-based methods for solving large-scale
differential Sylvester matrix equations having a low rank constant term. We
present two new approaches for solving such differential matrix equations. The
first approach is based on the integral expression of the exact solution and a
Krylov method for the computation of the exponential of a matrix times a block
of vectors. In the second approach, we first project the initial problem onto a
block (or extended block) Krylov subspace and get a low-dimensional
differential Sylvester matrix equation. The latter problem is then solved by
some integration numerical methods such as BDF or Rosenbrock method and the
obtained solution is used to build the low rank approximate solution of the
original problem. We give some new theoretical results such as a simple
expression of the residual norm and upper bounds for the norm of the error.
Some numerical experiments are given in order to compare the two approaches.
