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:
@Michael E2
Your suggestions were great. Here are the results.
Nice example by @David Slater in Plotting a Phase Portrait
Phase portraits
@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?
@yarchik: Cleaner Approach to Plotting Multiple Solution Curves on Phase Portrait
@Artes, @Sektor, @Michael E2: Phase portraits and StreamPlot
@Rahul: Why isn't my Stream code plotting multiple solution curves?
@ubpdqn: Posted to this question
Direction Field
@Jens: Plotting the direction field of a differential equation
@Jens: Graphics Grid Direction Field Animations as GIF
@Dr. belisarius, @Wesley Wolfe: How can I plot the direction field for a differential equation?
dfield
orpplane
would likely find this question and the linked solutions. $\endgroup$