Name_______________________________    UTEID_____________

Mathematics 427k       Uhlenbeck       Due March 10


1.  Use an algebraic method  (a solver or your own algebra)  to write
the solution of the equation

          y'' +  4 y  =  4 cos (2.2 t)

with y(0) = 0 and y'(0) = 0  as a product of two sine functions.









2.  Use an ode solver to graph the solution that you produced for the first
problem. Use a scale which includes at least two periods of the smaller
frequency. What are the angular frequencies of the sine functions?



3.  Use pplane and experiment to find the solutions of the system

       x' =  2x + 3y
       y' =  4x - 2y

which lie on  straight lines in the x, y plane. You do not have to solve
the equations.  Just demonstrate that some of the solutions lie on
straight lines and give the (approximate) equation of the lines.








                         3  -2
4.  Let A be the matrix (2  -2).   Find  A, A transpose, det A  and A^-1.
                                                          3-r  -2
Let I be the identity matrix and find det (A - rI)  = det(2    -2-r).













5. Let  A be as in problem 4.  Assume that the vector X =  exp(rt)4.
                                                           exp(rt)2
Find the constant r so that X' = AX.