Direction Fields and Euler's Method

Instructions:

The applet displays the direction field for the differential equation

dy/dx = ax^m + bxⁿ . You choose the parameters a, b, m, n, by using the sliders or by typing directly in the right-hand control panels. The applet draws the direction field.

The display is the region of the first quadrant with 0 < x < 3, and 0 < y < 3.

The applet redraws itself whenever you change the direction field parameters; this may take several seconds. It also redraws itself when you scroll the page.

The applet can use Euler's method to find and display approximate solution curves of the differential equation. You choose the step size h and the initial point for the solution curve; the applet iterates to the right of the initial point.

The applet should render properly using any Java-enabled browser, such as Netscape 3.0. Macintosh users with Netscape 3.0 may experience difficulties.

If you receive a message informing you that your browser "does not understand the applet tag," examine the Security Preferences Menu of your browser. There should be a checkbox which you can use to enable Java; absence of such a box probably indicates you are using an older web-browser and you must upgrade it in order to run the applet. Please take a moment to send me an e-mail reporting any difficulties you may have experienced with the applet.
Mac Users: This applet appears not to funciton properly under Netscape on the Mac. Try Apple's Applet Runner, Javasoft's Applet Viewer, or any other Java-enabled browser instead.

Questions to Investigate:

We restricted the applet display to the region 0 < x < 3 , 0 < y < 3 in the first quadrant. Why do you think we display only the first quadrant?
Some solution curves pass off the screen below the x-axis and then come back on-screen. Can you make a direction field that will produce this effect? If so, make some solution curves. Do the solution curves look correct? If not, what do you think is happening?
What should the solution curves look like in the case b = 0 ?
In the case n = 1, the differential equation assumes a special form What form? What should the solution curves look like in this case?
When a and b are both positive, do you think the approximate solution curves displayed by the applet underestimate or overestimate the true solutions? Why? [Hint: Vary the step size, keeping the other parameters constant.]
When the step size h is very small, the applet sometimes draws straight line solution curves even when such curves are obviously wrong. Can you explain this, using your knowledge of Euler's method and some knowledge of computers?
A particularly interesting case occurs when, for example, a = -b = 2 and m = n = 2. No matter what initial point you choose, all solution curves eventually approach and stay close to a particular curve. What is the equation of this curve? For what other values of the parameters does this phenomenon occur?