How do you solve a general second order matrix differential equation?

Question

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 $X = \begin{bmatrix}X1 \\ X2 \\ X3\end{bmatrix}$.

Answer 1

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.

Answer 2

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.