Name_____________________________       UTEID____________________

Math 427K           Uhlenbeck    Due April 28, 2005   Homework 10
Please answer the problems on this page and staple your work to the back of
the answer sheet.


1.  Use algebra and trig identities to find the Fourier series of the
function, which is  periodic  of period 2 pi, on the interval - pi <x < pi.

    f(x) =  3 - 5sin(x) cos(x) + 7(sin(x))^2.








2.Find the Fourier series on  the interval  -1 < x < 1  of the function
f(x) = 1 -3 x^2.






3.  Find the Fourier series   on the interval - 1 < x < 1 of the function 
f(x) = x - x^3.  Explain the relation between your answer to  problem 2 
and problem 3.









4. Carry out the method of separation of variables for the heat equation

    d/dt u  =  (d/dx)^2 u  on the interval  0< x < 4 with  (d/dt)u(t,0)= 0
and (d/dt)u(t,4) = 0. What is the general form for the solution?









5.  Reduce the partial differential equation
               d/dt u =(d/dx)^2 u  + 1/x (d/dx)u
to a pair of ordinary differential equation in t and x.  Here you need not
worry about the boundary conditions.