I wonder whether the matrix differential equation $$\dot x(t) + A(t) x(t)=0,$$ can be solved exactly. Here, $x(t)$ is a 4-component vector and $A$ is a $4\times 4$ matrix.
Question: Does the above equation exactly solvable, for general $A(t)$?
If not, suppose further that $A(t)$ has the following form $$A(t) = \begin{pmatrix} 0&c& e^{i\omega t}& 0 \\ c&0&0& e^{i\omega t} \\ e^{-i\omega t} & 0& 0& c \\ 0& e^{-i\omega t}& c& 0 \end{pmatrix}.$$ Here, $c, \omega$ are constants. With these assumptions, does the above differential equation exactly solvable?
If the answer is again no, then I wonder the exact solvability of the following simplified differential equation: $$\dot y(t) + B(t) y(t)=0,$$ where $$B(t) = \begin{pmatrix} 0 & e^{i\omega t} \\ e^{-i\omega t} & c \end{pmatrix}.$$ Again, $c,\omega$ are constants.
The above questions are motivated from physics. Thanks in advance.