How can I solve this system of linear different equations? Here's the system $$\frac d{dt} \begin{bmatrix} x \\ y \\ z \\ p_1 \\ p_2 \\ p_3\end{bmatrix} = \begin{bmatrix} 0 &amp; A \\ B &amp; 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ p_1 \\ p_2 \\ p_3\end{bmatrix}$$
where $\begin{bmatrix} 0 &amp; A \\ B &amp; 0 \end{bmatrix}$ is a block matrix and $A = \frac 1{mR^2}I_3$ and $B = mR^2\omega^2\begin{bmatrix} -2 &amp; 1 &amp; 1 \\ 1 &amp; -2 &amp; 1 \\ 1 &amp; 1 &amp; -2\end{bmatrix}$.  $m, R,$ and $\omega$ are all just constants.  $x,y,z,p_1, p_2$, and $p_3$ are all functions of $t$.

I have a feeling I need to use the matrix exponential, but my brain isn't working right now.
Let $E = \left[ \matrix{1 &amp; 1 &amp; 1\cr 1 &amp; 1 &amp; 1\cr 1 &amp; 1 &amp; 1\cr} \right]$.
With Maple's help, I get (if $C$ is your $6 \times 6$ matrix)

$$ \exp(tC) = \dfrac{1}{3} \left[\matrix{E &amp; (t/mR^2) E\cr 0 &amp; E\cr} \right] + 
\cos(\sqrt{3}\omega t) \left[ \matrix{I_3- E/3 &amp; 0\cr 0 &amp; I_3 - E/3\cr}\right] + \dfrac{\sin(\sqrt{3}\omega t)}{\sqrt{3}} \left[ \matrix{0 &amp; \dfrac{I_3 - E/3}{mR^2\omega}\cr mR^2\omega(E-3I_3) &amp; 0\cr} \right]$$
Let $E = \left[ \matrix{1 &amp; 1 &amp; 1\cr 1 &amp; 1 &amp; 1\cr 1 &amp; 1 &amp; 1\cr} \right]$.
With Maple's help, I get (if $C$ is your $6 \times 6$ matrix)

$$ \exp(tC) = \dfrac{1}{3} \left[\matrix{E &amp; (t/mR^2) E\cr 0 &amp; E\cr} \right] + 
\cos(\sqrt{3}\omega t) \left[ \matrix{I_3- E/3 &amp; 0\cr 0 &amp; I_3 - E/3\cr}\right] + \dfrac{\sin(\sqrt{3}\omega t)}{\sqrt{3}} \left[ \matrix{0 &amp; \dfrac{I_3 - E/3}{mR^2\omega}\cr mR^2\omega(E-3I_3) &amp; 0\cr} \right]$$