Chapter 8.1, Problem 26E

To determine

To prove: The general solution of the nonhomogeneous linear system $X^{\prime }=\left(\begin{array}{cc}-1& -1\\ -1& 1\end{array}\right)X+\left(\begin{array}{c}1\\ 1\end{array}\right)t^2+\left(\begin{array}{c}4\\ -6\end{array}\right)t+\left(\begin{array}{c}-1\\ 5\end{array}\right)$ on the interval $\left(-\infty ,\infty \right)$ is $X=c_1\left(\begin{array}{c}1\\ -1-\sqrt{2}\end{array}\right)e^{\sqrt{2}t}+c_2\left(\begin{array}{c}1\\ -1+\sqrt{2}\end{array}\right)e^{-\sqrt{2}t}+\left(\begin{array}{c}1\\ 0\end{array}\right)t^2+\left(\begin{array}{c}-2\\ 4\end{array}\right)t+\left(\begin{array}{c}1\\ 0\end{array}\right)$.