Math 255 Computer Project Part I - Due 10/2

This project is intended to make everyone familiar with some computer algebra package for solving differential equations. You may use any such algebra package that you wish (Maple, Mathematica, and Matlab are available in Dailey 201 and 202). You may also get a copy of either Maple or Mathematica for your own computer from ITS (FYI- these programs cost about $150 each the last time I checked). However, you must do your own work- identical assignments or ones that are almost identical will not be accepted. Make sure to label your solutions enough so that I can see what you are doing- and preferably include the commands that were used to create your solutions. Some information on how to do these calculations is located on the course website and more will be added as I have time. Click here to go to that page

(1) Direction fields:
    (a) First get the package to plot direction fields for three differential equations of your choosing. Pick a variety of first order diff. eqs (i.e perhaps one is linear, one is non-linear, one is autonomous, one is really ugly etc).
     (b) For each equation pick a couple of initial conditions and plot the solutions that meet the initial conditions- try to get very different solutions by picking the initial conditions appropriately.

(2) Solving differential equations:
    (a) First get the package to solve five different first order differential equations. Again, pick a variety of equations try some that you could solve and some that you could not. For at least three of them give it an initial condition as well. Plot graphs for these three solutions to initial value problems.
    (b) Solve four higher order equations (order at least 2) - use constant coefficients and homogenous, and have the program solve these equations. For two of them give the program initial conditions and ask for the specific solution. (note that for an equation of order n you need to specify n initial conditions, usually of the form y(x₀)=y₀ , y'(x₀)=y₁, y''(x₀)=y₂... y^(n-1)(x₀)=y_n-1  ). What different functions do you find arise as solutions to these equations?
    (c) For two of the above equations try a non-homogenous version (so instead of y⁽ⁿ⁾+a_n-1y^(n-1)+...+a₁y'+a₀y=0 you have y⁽ⁿ⁾+a_n-1y^(n-1)+...+a₁y'+a₀y= some function of t). For each equation try two functions on the right hand side of the equation and determine how the answer changed from the homogenous case.

If you are interested in particular types of equations these would be good examples to use above (for example a physicist might like pendulum equations, a chemist might like chemistry diff eqs, meteorology might like those equations etc. just look in your diff. eq text or the ones from your other classes to see what diff. eqs arise and are of interest).

If you have any questions do not hesitate to ask.