How do you solve a general second order matrix differential equation? How do you find solutions to the equation of the form :
\begin{equation}A\frac{d^2X}{dt^2} + B\frac{dX}{dt} + CX = 0 \end{equation}
where A,B,C are 3X3 positive definite and symmetric matrices with constant values, and 
$X = \begin{bmatrix}X1 \\ X2 \\ X3\end{bmatrix}$.
 A: Transform it to a system:
$$ D\dot{X}+E\cdot X =0$$
Done the same way as for univariate DE. Elsewhere google.
This system then can be solved using standard methods.
A: I would suggest to convert the system to the standard form of linear 6-dimensional equation. We can do this in the following way:
$$
Y_1=\left(\matrix{X1 \\
          X2  \\
          X3 }\right),~ Y_2=\dot Y_1=\left(\matrix{\dot X1 \\
          \dot X2  \\
          \dot X3 }\right ), ~
Y=\left(\matrix {Y_1\\
                Y_2}\right)= 
\left(\matrix{X1 \\
          X2  \\
          X3 \\
          \dot X1 \\
          \dot X2 \\
          \dot X3 \\ 
}\right)\\~ 
A \dot Y_2 +\tilde B Y_2+CY_1=0\\
\dot Y_1 =Y_2
$$ 
So we have
$$
\tilde A \dot Y +\tilde B Y=0
$$
where 
$$
\tilde A =\left(\matrix{I & 0 \\
                        0 & A}\right )\\
\tilde B=\left(\matrix{ 0 & -I \\
                        C & B}\right )
$$
Now if $A$ is not singular you can rewrite the last equation as 
$$
\dot Y=\tilde A^{-1}\tilde B Y
$$ 
This is the standard from of linear equations. If $A$ is is singular the situation is more complicated. We should convert $A$ to eigen basis.  Part of differential equation  will be reduced to algebaric.
