Diagonalization & Matrix Powers · Hard problem

The equation $y^{\prime\prime\prime} - 2y'' - y' + 2y = 0$ is the system $\mathbf{u}' = C\mathbf{u}$ for $\mathbf{u} = (y, y', y'')$ with the companion matrix $C = \begin{pmatrix}0 & 1 & 0\\0 & 0 & 1\\-2 & 1 & 2\end{pmatrix}$. Using the eigenvalues of $C$, solve with $y(0) = 0$, $y'(0) = 0$, $y''(0) = 6$, and find $y(\ln 2)$.