The ti89 can almost graph vector fields, all two things are missing,
the arrowheads and all vectors are all the same size.

0. Turn on the calculator.
1. Use the mode button; select graph; select #6 diff equations.
2. green diamond y=; green diamond `|' (near 7); fields (near bottom); #2 dirfld
3. green diamond y=; Use y1 for x and y2 for y. [So if F = <-y, x> then
the flow equations are x' = -y and y' = x. One must enter y1' = -y2 and
y2' = y1 in ti89 y= window.]

The initial values will give trajectories. There are two methods Euler
and RK. Euler is easy to describe, but crude; RK is more accurate. You can
select between them via
 green diamond y=; green diamond `|' solution method (near bottom);
Trajectories are covered in HH17.4 where she calls them flows.

To get the values used by the program to plot the trajectories can
be obtained via the table interface. Assume you have done 1-3 above
and continue:
4. green diamond F4 (TblSet). TblStart should be 0 and Delta tbl should
match the tstep in the Window.
5. green diamond F5 (TABLE).

One should now have a table with 
t    y1 (really x)  y2 (really y)
0    initial x      initial y
.1   next x         next y
and so on.