Chapter 9.4, Problem 25E

Verify that the vector functions

$x_1=\left[\begin{array}{c}{e}^t\\ e^t\end{array}\right]$ and $x_2=\left[\begin{array}{c}{e}^{-t}\\ 3e^{-t}\end{array}\right]$

are solutions to the homogeneous system

$x^{\prime }=Ax=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]x$

On $\left(-\infty ,\infty \right)$, and that

$x_p=\frac{3}{2}\left[\begin{array}{c}t{e}^t\\ t{e}^t\end{array}\right]-\frac{1}{4}\left[\begin{array}{c}{e}^t\\ 3e^t\end{array}\right]+\left[\begin{array}{c}t\\ 2t\end{array}\right]-\left[\begin{array}{c}0\\ 1\end{array}\right]$

Is a particular solution to the nonhomogeneous system

$x^{\prime }=Ax+f\left(t\right)$, where $f\left(t\right)=\text{col}\left(e^t,t\right)$. Find a general solution to $x^{\prime }=Ax+f\left(t\right)$.