Maple Assignment 1


Use the Maplework sheet titled: Numerical Methods Problem to analyze the ODE:

y' = x - y^2.

Please complete the following: (Note: You can re-use the given worksheet but you must change the definition of the function in the worksheet for this problem.)

  1. Generate a direction field plot for this ODE using the DEplot command.
  2. Sketch (by hand in red ink) on the direction field plot, the trajectory of the solution with initial condition y(0)=1 . This is a very rough sketch of the solution trajectory.
  3. Follow the worksheet to generate approximate solutions using the Euler and Improved Euler Algorithm as well as the exact solution. The exact solution is complicated, involving Bessel functions, and Maple uses a bit of time to construct it. Be sure to label the curves in your graph indicating what method was used to generate it.
You are required to submit both graphs with your Maple Worksheet. Your name should be in the title of each graph in the Maple command. Be sure to label the graphs for each method. In a sentence or two, identify which numerical method gives the best approximation for x large and give a reason why. Suppose you generated the solution for a longer time, would your conclusion be different.

Remember, you must submit 2 graphs and an explanation for grading!
Notes:
  1. Maple Worksheet Euler.ms is MapleV Release 3, so if you are using MapleV Release 5, as you should be using, you must replace   rhs(")   by   rhs(%)   in the 4th line from the bottom (Waterloo Maple Software made this downwardly incompatible change in Release 5).
  2. If you are having trouble with the Worksheet Euler.ms popping up when you click on the URL Link, e.g., with Macintosh's, then try saving Euler.ms in a file and opening it directly from Maple.
Updated 25 Jan 99 --- http://www.math.uic.edu/~hanson/math220/maple_assign.html