NAME____________________________    UTEID____________________


Math 427K  Uhlenbeck    Homework 1  Due 1/27

Please write your answer on this page.  Include your computations
on extra sheets if neceessary. The notation a^k means a to the k-th power.

1. circle the letters of the equation which are linear equations.

a) y' =  3y + 5.                  b) y' = ty - 3     c) y' = sin(t)y

d) y"  = y'y                      e) y =  y^2        f) y" = t y' - t^3.

2.  a) Find the solution to   y' + .1 y - 5  = 0 with y(0) = 0.




    b)  At what time does y = 25?




    c)  What is the limit of the solution y(t) as t goes to infinity?



3.  Find the number r such that  y = exp(r t)  solves each equation.

    a) y' + 5y = 0.                 b) y" - 2y' + y  = 0.
 




    c) y" - 4y' + 3y = 0.           d)  y" + 4 y = 0.
 




4.  Find the general solution to the equation  y' +  y  = cos(2t).





5.  A population of mosquitoes doubles every week.  At the beginning of
measurements, there are 1 million mosquitoes.  Frogas are introduced
which eat mosquitoes at the rate of r  mosquitoes per week. How
big must r be in order to insure that the mosquito population is
destroyed?