Direction of arrows in a phase portrait. If I have a system like: $\frac{dx}{dt} = x, \frac{dy}{dt} = -y+x^2$ and I am asked to draw a phase portrait, once I have found the type of portrait (saddle point, node, spiral, etc.) from the eigenvalues and have found the $\infty$-isocline and $0$-isocline, how do I determine the direction of the arrows on the portrait?

I assume that the second equation should start with $\frac{dy}{dt}\equiv \dot y$.

An easy way to figure out the directions is to put arrows on $0$-isoclines, which are vertical for $\dot x=0$ and horizontal for $\dot y=0$. To determine on which end to put an arrow you simply check the sign. For example, if $\dot y>0$ then your arrow on the $x$ isocline is $\uparrow$. Otherwise it is $\downarrow$. The directions on other parts of the portrait follow by continuity.