Section 3. Slope Fields: DEplot

The DEplot procedure draws slope fields and solution curves directly from the ode.

0. Pull down the Help menu, choose New Users/Full Tour, then click on the item eight: Differential Equations and work through the ordinary differential equation examples in that worksheet paying special attention to the section on the DEtools package.

Open a new Maple worksheet and do the following.

1. Consider the following first order differential equation taken from Example 1 in Section 2.3 of Ledder.

(t+1)*(diff(y(t), t)) = 1-y(t)

a. Use dsolve to obtain the general solution and also the solution satisfying y(0) = 0. Plot the second solution using t=-2..2, -2..2 and compare the picture to the solution curve shown in Figure 2.3.4 in Ledder and reproduced below.

[Plot]

b. Attempt to reproduce the picture above using DEplot.

c. Get a better picture by doing the following. Draw the six solution curves in one plot using the phi function for the differential equation. Immediately after making a plot you like, use a DEplot entry to make the slope field (no solution curves), but terminate the entry with a colon. Then make the following entry

plots[display]( %, %% );

d. Comment on the existence and uniqueness of solutions using the statement of the Unique Solution Theorem. Your comments should be based upon the properties of the function  f (t,y) where  y' = f (t,y).

2.* Use DEplot to make a nice-looking slope field for the autonomous equation y' = sin(y). (Use the window t=-6..6, y=-6..6.)

a. Put some solution curves into the plot by adding { [y(0)=k] $ k=-6..6 }, linecolor=blue . Comment on the relationship between one curve and the next.

b. Obtain the general solution using dsolve.

c. Use dsolve to obtain the solution satisfying the initial condition y(0) = 1. What is the value of this solution when t = 1? Get the exact value and an approximation. If the solution is called "soln" use

eval(soln,t=1); evalf(%);

d. Comment on the long-term behavior of solutions to this differential equation.

3. Repeat Exercise 3 using the differential equation y' = 1/sin(y). Use the same plot window and inits, but reduce the stepsize to 0.1. Then make two more DEplots in the window t=0..3, y=0..4 with only two inits, y(0)=1 and y(0)=2 and stepsize 0.1. First use the default numerical method, then use the optional Runge-Kutta Fehlberg method by adding the equation method=rkf45. (Copy and Paste to make the new entry.)

Note that the symbolic solutions are much simpler than the ones in Exercise 3. Explain why.