As a follow up to a question I asked yesterday, the following type of differential equation also started to appear in some gauge theory calculations that I've been doing recently $$\text{d}U=-(U\cdot\Omega^\vee_1+\Omega^\vee_2\cdot U).$$ Here, $U$ is a $n\times n$ matrix of functions (in many variables) I would like to solve for, while both $\Omega^\vee_{1,2}$ are $n\times n$ matrices of 1-forms (in the same many variables) I already know from previous calculations.
Again, I would (ultimately) like to know if a general solution to such DE is known; so any tips from experts or references from the math/physics literature would be really appreciated.
Thanks.
It appears that the answer is easy to derive; I just tried the following natural ansatz and it worked
$$U(\textbf{s})=\left(\mathcal{P}\exp\int \Omega_2^\vee\right)\cdot U(\textbf{s}_0)\cdot\left(\mathcal{P}\exp\int \Omega_1^\vee\right),$$
where $\mathcal{P}$ stands for usual path-ordered exponential (one acting from the left and the other from the right) and where $\textbf{s}$ is the collection of active variables.