Question $5.$ Match the following differential equations with the corresponding phase portraits:

Equations:

Phase portraits:

$($i$)$

${\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]}^{\text{'}}=\left[\begin{array}{cc}1& -1\\ 1& 3\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

$($ii$)$

${\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]}^{\text{'}}=\left[\begin{array}{cc}\frac{-1}{2}& 1\\ -1& \frac{-1}{2}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

$($iii$)$

${\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]}^{\text{'}}=\left[\begin{array}{cc}1& 1\\ 4& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

$($iv$)$

${\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]}^{\text{'}}=\left[\begin{array}{cc}-3& \sqrt[\phantom{2}]{2}\\ \sqrt[\phantom{2}]{2}& -2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

$($a$)$

$($b$)$

$($c$)$

$($d$)$