I am analyzing the stability of the following differential equation system: \begin{equation} \left[\begin{array}{c} \dot{x}_{1}\\ \dot{x}_{2} \end{array}\right]=\left[\begin{array}{cc} -1 & 1\\ -1 & 1.5 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]+\left[\begin{array}{c} 9\\ 9 \end{array}\right]. \end{equation} The eigenvalues of the coefficient matrix are given as $-0.5$ and $1.$ Hence, the critical point given as $(9,0)$ is unstable.
Given the problem specific (initial) conditions that $x_1(0)=0$ and $\lim_{t\rightarrow\inf}x_1(t)x_2(t)=0$, I derive the solution as \begin{align} x_{1}\left(t\right)&=9\left(1-e^{-\frac{t}{2}}\right)\\x_{2}\left(t\right)&=-4.5e^{-\frac{t}{2}}. \end{align}
As $t$ tends to infinity, $x_1$ approaches to $9$, whereas $x_2$ approaches to $0.$ So, it seems to me that $(9,0)$ is a stable equilibrium.
I am not sure how a system with nonnegative eigenvalues would lead to a stable solution.