Consider this System:
$$ X'= \begin{pmatrix} 1 & -5 \\ 1 & -3 \end{pmatrix} X $$ and $$ X= \begin{pmatrix} x\\ y \end{pmatrix} $$
I want to get an approximation of phase plot (around $(0,0)$) and plot of each $x$ and $y$ in terms of $t$. note that I don't want to use software or any calculator and I want to use easy mathematical and numerical ways to approximate the shape.
For eigenvalues, I got $\lambda =\pm i - 1 $. because of $-1$ the $(0,0)$ point is asymptotically stable and a spiral point. For drawing $y_x$ plot (phase plane), I just plot some point on the equation and found some X' and drawed that with the direction towards $(0,0)$. But I don't know how to plot $x$ and $y$ in terms of $t$ (time). What should I do?