I have thus far mostly (only) been dealing with phase portraits where you compute eigenvalues and eigenvectors and from there are able to sketch the portrait. Now I came across a different type of example, and I haven't been able to find in my literature any real explanation as to how one approaches problems of this kind.
For the system $$ \left\{ \begin{array}{c} \frac{dx}{dt} = &ye^y \\ \frac{dx}{dt} = &xe^{x+y} , \end{array} \right. $$
I'd like to sketch the phase portrait. I have been playing around with it a little, trying to sort of guess my way through the problem. But I could really use some pointers and advice.
So first of all, we have $\frac{dy}{dx} = xe^x$, giving the level curves $\frac{y^2}{2} = (x+1)e^x + C$. So the first thing I did was trying to sketch some of these curves for different constants. And then I tried to reason a little bit about what happens depending on whether $x(t)$ and $y(t)$ are positive or negative. So for example, if $x(t) < 0$ and $y(t) > 0$, then $x' >0$, so $x(t)$ is increasing, while we have $y'<0$, so $y(t)$ is decreasing. I figured this could help with the direction of the arrows in the sketch.
Also, the only critical point seems to be at the origin, but I'm not sure what to do about that information.
I'm unsure of whether this really is a good way to solve these types of problems. Sketching level curves seems really time consuming and depending on the equations of the curves it could get quite complicated. Is there some other general method that I'm not aware of?