Assignment 4  Due February 26    Computer Assignment


This is an assignment to use three computer programs. If you do not know
a computer program that takes roots of polynomials, go to the help section
of matlab (or whatever you are using).

pplane is a program of matlab very similar to dfield.  To use it, you must
convert the second order equation into a system.

ode45 is another program of matlab...you can, of course, use any ode solver
that will give you output as graphs.  There is a help sheet on ode45 on the
home page for the course.

1.Use a computer program to find the roots of the equation 

r^4 +5r^3 +7r^2 +5r +1.  Now give the general solution to the equation

y'''' + 5y''' + 7y'' + 5y' + y = 0.   


2.Consider the equation x" + 5x'- 6x = 0.  Use pplane to make a graph of
the x(t), x'(t)  curve in phase space.  Plot a few typical solutions.
What happens to most solutions as t goes to + infinity? Be as specific
as you can be.


3.  Consider the equation x" + 5x' + 4x = 0. Use pplane to make a graph.
What happens to most solutions as t goes to infinity?  Where do solutions come
from if you look back to - infinity?


4. Make a meaningful ode45 plot of a  solution to the equation

                  x" +  x  =  sin(1.2t).

(Meaningful means that you have set the time limits and chosen fairly typical
initial conditionsappropriately so that you see what is happenning).
 


5.A  basic preditor-prey system  for the population of preditors N
and prey M  has the form (the constants are somewhat arbitrary)

                     M' = M(5 - M/10  - N)
                     N' = 2N(-1 + M/10).

Make both a pplane plot and an ode45 plot of the solution with M(0) = 1,
N(0) = 1.