Please show us what you have tried and exactly where you are stuck.
Here's a potential starting point. Ideally what we would like to do is to just let [math]u = \left ( \begin{matrix} x(t) \\ y(t) \\ z(t) \end{matrix} \right )[/math] and multiply it out to get three uncoupled equations, but the way it stands the matrix A is going to mix the functions x(t), y(t), z(t) in each of the three equations: like [math]\dfrac{du}{dx} = ax(t) + by(t) + cz(t) + \text{ exponential terms.}[/math] So we need to diagonalize A (aka change the basis) so that when we set [math]u = \left ( \begin{matrix} x(t) \\ y(t) \\ z(t) \end{matrix} \right )[/math] we get three equations in x(t), y(t), and z(t) that aren't mixed together and then you can do undetermined coefficients with each equation. (Or all at once if you prefer to keep working with the matrices.)