Ordinary Differential Equation System with Variable Coefficients

[cathode-ray]

Homework Statement

For $$t\in\mathbb{R}$$, let:

$$A=\left[\begin{array}{ccc}<br /> -1 & 0 & 0\\<br /> 0 & 2 & 2t\\<br /> 0 & 0 & 2\end{array}\right]$$

Get the solution for the general equation: $$X'=A(t)X$$

Homework Equations

The Attempt at a Solution

I done many of these problems, all with constant coefficients, but I don't know how to do in this case.

 

[CompuChip]

Did you know that the solution to
$$x'(t) = f(t) x(t)$$
is
$$x(t) = x_0 \exp\left( \int_{t_0}^t f(\xi) \, d\xi \right)$$
?

 

[cathode-ray]

Yes, I know. That's the formula I use after getting the exponential matrix, by "diagonalizing" the matrix A. My problem is that I'm not sure if I can do it as I do with constant coefficients, because supposedly it should be different.