Suppose I have a system of ODEs $\dot{x} = (At+B)x$, where $x(t)$ is a, say, $n \times 1$ vector, and $A$ and $B$ are constant $n \times n$ matrices. What is the fundamental matrix of this system?
I tried $e^{\frac{At^2}{2}}e^{Bt}$, trying to emulate the one-dimensional solution, but then A and B don't necessarily commute for it to work. Do I need the Lie bracket $[A,B] = AB - BA$ somewhere? Do I need something like power series?