Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients, $$Y(t_1) = \exp(\Omega(t_1,0))Y(0),$$ or, in a more general setting, $$Y(t_1) = \exp(\Omega(t_1,0))Y(0)+c(t_1,0),$$ where the $\Omega(t_1,0)$ is the Magnus expansion just as in https://en.wikipedia.org/wiki/Magnus_expansion and http://arxiv.org/pdf/0810.5488.pdf .

If I assume that the system of linear ODEs is given by, $$\frac{dY}{dt} = A(t)Y(t)$$ or, more generally, $$\frac{dY}{dt} = A(t)Y(t)+b(t),$$ is there any way to get out some information about $A(t)$ or $\int_0^{t_1} A(t) dt$ or $b(t)$ by just knowing $\Omega(t_1,0))$ (and $c(t_1,0)$).