The toolkits dfield and pplane are staples in ODE courses. First in MATLAB, now in Java. Do they have a Mathematica expression?
Sample outputs follow:
Your suggestions were great. Here are the results.
Nice example by @David Slater in Plotting a Phase Portrait
@Chris Degnen: Phase Portrait Trajectories
@chris: Plotting a Phase Portrait
@Kuba♦, @Jens: Phase portrait on a cylinder
@Alexei Boulbitch: How to make program for phase portrait?
@Artes, @Sektor, @Michael E2: Phase portraits and StreamPlot
@ubpdqn: Posted to this question
@Dr. belisarius, @Wesley Wolfe: How can I plot the direction field for a differential equation?