Real Fundamental- System/Matrix of a Differential equation We consider: 
$$y''' - 2y'' + 2y' - y = 0$$
The real solution to this equation is: 
$$y(x) = c_3e^{x} + c_2e^{x/2}sin\left(\frac{\sqrt{3}x}{2}\right) + c_1e^{x/2}cos\left(\frac{\sqrt{3}x}{2}\right)$$
How do we now represent it as a fundamental- system/matrix ? 

Write your DEQ as a System of First Order Equations, find eigenvalues / eigenvectors and proceed in the usual way.
We have $$y''' - 2y'' + 2y' - y = 0$$
To write it as a system of first order equations we let $x_1 = y$, so
$$\begin{align} x_1 ' &= y' = x_2 \\ x_2' &= y'' = x_3 \\ x_3' &= y''' = 2y'' - 2 y' + y = 2 x_3 - 2 x_2 + x_1 \end{align}$$
In matrix form, we have
$$X' = Ax = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & -2 & 2 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$
Can you proceed?
