Vector fields

The vector field for a system of differential equations (first order, autonomous) is pretty easy to plot using PlotVectorField:

(Notice, you don't have to actually solve the system to plot this.)  If you want to plot the "direction field", you just need to arrange that all the vectors have the same length:

Notice that these are plots in the XY plane (i.e., time doesn't enter into this).  So, what would it mean to plot solution curves on this plane?  It would mean to graph the parametric equations:  , graphing x vs. y in the plane (the "trajectory" in the XY plane).

If , then we have:

To show a bunch of curves: