System of equations:

$$\left(\begin{array}{c}{\dot{y}}_{1}\\ {\dot{y}}_{2}\end{array}\right)=\left(\begin{array}{cc}2& 0\\ 4& -1\end{array}\right)\left(\begin{array}{c}{y}_{1}\\ {y}_{2}\end{array}\right)$$

Looking at matrix $A$ I can see a value not on the diagonal equal to zero. So the eigenvalues here are $2$ and $-1$, meaning I have a saddle node. Next I look at the eigenvectors:

$\lambda =2,\left[\begin{array}{ccc}0& 0& 0\\ 4& -3& 0\end{array}\right]$

$\lambda =-1,\left[\begin{array}{ccc}3& 0& 0\\ 4& 0& 0\end{array}\right]$

$y=A{e}^{2x}\left(\begin{array}{c}3\\ 4\end{array}\right)+B{e}^{-x}\left(\begin{array}{c}0\\ 1\end{array}\right)$

So I have an attractive trajectory on the $y$-axis, and a repulsive trajectory on the line $y=\frac{3}{4}x$

With the slope at origin being

$\frac{{\dot{y}}_{2}}{{\dot{y}}_{1}}=\frac{4{y}_{1}-{y}_{2}}{2{y}_{1}}=\frac{0}{0}=\pm \mathrm{\infty}$

Does all of this seem correct?

$$\left(\begin{array}{c}{\dot{y}}_{1}\\ {\dot{y}}_{2}\end{array}\right)=\left(\begin{array}{cc}2& 0\\ 4& -1\end{array}\right)\left(\begin{array}{c}{y}_{1}\\ {y}_{2}\end{array}\right)$$

Looking at matrix $A$ I can see a value not on the diagonal equal to zero. So the eigenvalues here are $2$ and $-1$, meaning I have a saddle node. Next I look at the eigenvectors:

$\lambda =2,\left[\begin{array}{ccc}0& 0& 0\\ 4& -3& 0\end{array}\right]$

$\lambda =-1,\left[\begin{array}{ccc}3& 0& 0\\ 4& 0& 0\end{array}\right]$

$y=A{e}^{2x}\left(\begin{array}{c}3\\ 4\end{array}\right)+B{e}^{-x}\left(\begin{array}{c}0\\ 1\end{array}\right)$

So I have an attractive trajectory on the $y$-axis, and a repulsive trajectory on the line $y=\frac{3}{4}x$

With the slope at origin being

$\frac{{\dot{y}}_{2}}{{\dot{y}}_{1}}=\frac{4{y}_{1}-{y}_{2}}{2{y}_{1}}=\frac{0}{0}=\pm \mathrm{\infty}$

Does all of this seem correct?