Homewor 4   Due February 23



1.  Write the following complex numbers in polar form and mark them
on the complex plane (graph paper would be best, but lined paper will
do).

a)  1.5 i       b) 1 + i      c)  3^(1/2) + i   d) (2 - i)(2 + i)



2.  Write the follow complex numbers in Cartesian form

a) exp(-i\pi)  b) exp(-1+i(\pi/2)) c) exp(-1(\pi)/4) d) 2^(i(\pi)/3)

3. Find the general solution of the linear equation

     y" + 6y' +10y = 0.


4. Find the solution of the differential equation


     y" - 2y' + 5y = 0

with y(0) = 0 and y'(0) = 1.

5.  Use what you know about linear equations to solve the equation

     y" + 4y = 4  with y(0) = 0 and y'(0) = 0.



Extra Credit:


6.  Prove that if p is a prime, then every element  z  in  Z/pZ has
an inverse.

7.  Describe the solutions to the equation y" + by' + y = 0 as a function
of the parameter 0<b< infinity.  Find the value of b at which a bifurcation
occurs and describe in words what happens at this point. (It is difficult
to draw, unfortunately).