### Maple Tutor

Part 12: Direction fields

1. Enter the command
with (DEtools);
to load a package of programs -- differential equation tools. The only one of these tools we will use here is "dfieldplot." To see a direction field plot for the logistic differential equation in Part 11, enter
dfieldplot(DE1, y, t = -2..2, y = -0.5..1.5);
The first entry in parentheses is the differential equation. The second entry names the dependent variable. The third and fourth entries give ranges for the independent and dependent variables.

2. Now plot the function h(t) from Part 11 with the same limits on t and y. Does this solution look like it fits the direction field?

3. In order to be sure the symbolic solution fits, we need to overlay the two pictures being drawn by different plot routines. To do this, we assign each plot a name, and then we "display" a list consisting of the two names. To get the display command (and other graphing tools), enter
with(plots);
to obtain access to the display command. Then go back to your dfieldplot line and change it to
plot1 := dfieldplot(DE1, y, t = -2..2, y = -0.5..1.5, color=blue): %;

Recall the use of a colon to keep from outputting Maple's complete description of the plot. The %; at the end causes drawing to take place.

Change your function plot command to
plot2 := plot (h(t), t = -2..2, y = -0.5..1.5, thickness=2): %;
Then enter
display( plot1, plot2 );
to show the two plots superimposed.

4. Adjust the horizontal ranges in the preceding step so you can see the solution function h(t) approach equilibrium. If necessary, enlarge the picture so you can see more detail. Are you convinced that the symbolic solution h(t) fits the direction field?

5. Generate direction fields over appropriate ranges of the variables for the following differential equations
• dy/dt = y2

• dy/dt = ty + t