Chapter 6.3, Problem 27PS

To determine

The solution $x\left(t\right),y\left(t\right)$ that gets large as $t\to \infty$ and the reason how it become stable by just changing the order of derivative in the following way

   $\begin{array}{c}\frac{dx}{dt}=0x-4y\\ \frac{dy}{dt}=-2x+2y\end{array}\text{becomes}\begin{array}{c}\frac{dy}{dt}=-2x+2y\\ \frac{dx}{dt}=0x-4y\end{array}$

With coefficient matrix $\left[\begin{array}{cc}-2& 2\\ 0& -4\end{array}\right]$, which is stable because of negative eigenvalues.