I have been having difficulties finding the direction:
The system of ODEs is:
$$ \begin{cases} x' = -x, & x(0) =x_0\\ y'=-2y, & y(0) = y_0\\ \end{cases} $$
which I know its direction can be sketched as $y(x) = cx^2$.
I've plotted it online, and all arrows point towards (inwards) the origin (since $(0,0)$ is the equilibrium point), but for example, for $x,y>0$ then there would be some arrows pointing outwards (when computing the derivative). Can anyone illustrate the process? Or let me know what I don't understand.
The general solution is $x=x_0e^{-t}$, $y=y_0e^{-2t}$. The absolute values of both decreases as $t$ increases, so the arrows point towards the origin. There’s not really much more to it than that. Eliminating $t$ loses this information, which is why you can’t tell from the implicit trajectory equation $y_0x^2=x_0^2y$ which way the arrows should point.