I have the system of $m$ coupled ODEs given as:
$$A_4 x^{(4)} + A_3 x^{(3)} + A_2 \ddot{x} + A_1\dot{x}+A_0x =0$$
along with $4m$ initial (i) and final (f) conditions $x_i$, $\dot{x}_i$, $x_f$, and $\dot{x}_f$.
The coefficient matrices $A_0,\dots,A_4$ contain constant coefficients and they have some nice structure. The question is: what is some reference on solving this system directly without transforming it to 1st-order (if possible) and with getting some relationship between the structure of $A_0,\dots,A_4$ and the eigenvalues and eigenvectors?