  The numerical integration of stiff equations is a challenging problem that
needs to be approached by specialized numerical methods. Exponential
integrators form a popular class of such methods since they are provably robust
to stiffness and have been successfully applied to a variety of problems. The
dynamical low- \rank approximation is a recent technique for solving
high-dimensional differential equations by means of low-rank approximations.
However, the domain is lacking numerical methods for stiff equations since
existing methods are either not robust-to-stiffness or have unreasonably large
hidden constants. In this paper, we focus on solving large-scale stiff matrix
differential equations with a Sylvester-like structure, that admit good
low-rank approximations. We propose two new methods that have good convergence
properties, small memory footprint and that are fast to compute. The
theoretical analysis shows that the new methods have order one and two,
respectively. We also propose a practical implementation based on Krylov
techniques. The approximation error is analyzed, leading to a priori error
bounds and, therefore, a mean for choosing the size of the Krylov space.
Numerical experiments are performed on several examples, confirming the theory
and showing good speedup in comparison to existing techniques.
