While the technique described in this post works, there is a big-time shortcut to this problem that I initially oversaw. The shortcut works when A is a constant matrix. Check the posts after this one for an explanation of this shortcut method.
The idea of variation of parameters is to seek the solution to
in the form
whereby is a fundamental matrix.
Note that the constant vector c of coefficients is replaced by an unknown function v(t).
What is the equation for our unknown function v(t) you ask?
It is fairly straight forward to derive:
IMPORTANT: You must make sure that exists in order to solve for v !!!! (not factorial)
Finally, using integration, we have:
I'll now rewrite the original problem in latex so it is more palatable to the eye:
Find the general solution of, whereby .
Does this help?