How to draw a phase portrait of a two-dimensional ODE? If we're given:
$$\dot{x}=-x+y$$
$$\dot{y}=xy-1$$
How do I draw a phase portrait of this system? I don't understand which direction the arrows are supposed to point.
This is what I got so far:
I found the nullclines:
$$\dot{x}=0$$
$$x=y$$
and
$$\dot{y}=0$$
$$xy=1$$
Then I drew the lines  $x=y$ and $xy=1$. 
Don't know what to do from here.  
 A: This three step process is a summary from the excellent book series "Differential Equations: A Dynamical Systems Approach, Higher-Dimensional Systems" by Hubbard and West.


*

*$(1)$ Write the equations as:
$$\dfrac{dy}{dx} = \dfrac{\frac{dy}{dt}}{\frac{dx}{dt}} = \dfrac{xy - 1}{y - x}$$


and sketch the isoclines of $(a)$ Horizontal Slope (where $y' = 0$) and $(b)$ Vertical Slope (where $x' = 0$).


*

*$(2)$ In each region determined by these isoclines, put together the horizontal and vertical arrows and then sketch the resultant direction field. 


Here you are using the above equation, choosing sample $(x, y)$ pairs and drawing the arrows that have direction and magnitude based on the slope.


*

*$(3)$ Trace sample trajectories through the direction field.


Some other things that I find helpful are to determine the type of critical points which you see at the intersection of the nullclines, that is $(-1, -1)$ and $(1, 1)$ in this example. One is a stable spiral at $(-1, -1)$ and the other is an unstable saddle point $(1, 1)$. Additionally, you can look at the eigenvectors.
Putting all of these things together, we arrive at the phase portrait:

Lastly, it is worth noting, that these are not hard rules. Practice makes perfect and you'll develop your own approach in a way that is easy for you.
A: Note that your ODE is of the form $(\dot{x},\dot{y})=(F_1(x,y),F_2(x,y))$ and so you can plot the vector field $$(x,y)\mapsto(F_1(x,y),F_2(x,y)).$$
