Name_____________________________       UTEID____________________

Math 427K           Uhlenbeck    Due April 14, 2005

Please answer the questions on the problem sheet and attach your work on
the back.
 

                                     _     _
1.  Classify the critical point 0 of x' = Ax where
             3 -2                          2  5
        A =  4 -1,                    A  = 1 -2.









2.  Convert the spring equation y" + 2y'  + sy = 0 to a system, and classify
the nature of 0 as a critical point as a function of s.   









3.  Find and classify  the critical points of the system

    x' = -x  - y -  1;    y' = 2x - y + 5.  








4. Find and classify the critical points (x,y) = (a,b) where both a and b
are non-negative.

      x' = x(1 - x - 10y)
      y' = -y(1 - .5x).
 








5. Find and classify all the critical points of the system

x' = x(2  - x  - y);  y' = y(3 - y  - 2x)