Math 427K-H  Assignment 2   Due Feb 3, 2010



1.  Find the general solution of

       y' = -.5*y  + exp(-2*t)  -  2.

Print out a dfield plot showing the trajectory with y(0) = 10.



2.    Find the solution of  y' = -y + cos(t) with y(0) = 0. Print out
a dfield plot showing this solution.



3.    Consider the equation N' =  N (N - 1)(N - 20).  Give the fixed points
and decide whether they are stable or unstable.  Print out a dfield plot
which shows the behavior of the sample  solutions with y > 0.


4.  Give the interval of existence of the solutions to the following initial
value problems:

y' = y^3  with y(0) = 0;  y'= y^3  with y(0) = 1.  y' = t^3* y with y(0) = 1.

5.  Given  y' =  -2*y  + 3*H(t)where H(t) is the Heaviside function discussed
in class.  Are solutions continuous?  Demonstrate by solving the equation
with y(-1) = 1.

Remark:  The Heavyside function is the function
                    0   t< 0
             H(t) = 1/2 t= 0
                    1   t> 0.
Does it matter what H(0) is? Does it influence the solution?
 

Extra Credit:(Due Feb 4)  

6.Solve the initial value problem  y' + y = exp(-rt) and y(0) = 0 for
all r.  Show that the solution is continuous as a function of r. Pay
particular attention to r = 1.