Chapter 8.1, Problem 25E

Prove that the general solution of the homogeneous linear system

$x^{\prime }=\left(\begin{array}{ccc}0& 6& 0\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)x$

on the interval (−∞, ∞) is

$x=c_1\left(\begin{array}{c}6\\ -1\\ -5\end{array}\right)e^{-t}+c_2\left(\begin{array}{c}-3\\ 1\\ 1\end{array}\right)e^{-2t}+c_3\left(\begin{array}{c}2\\ 1\\ 1\end{array}\right)e^{3t}$.