Name_____________________________       UTEID____________________

Math 427K           Uhlenbeck    Due May 5, 2005   Homework 11
Please answer the problems on this page and staple your work and the computer
output for problems 4 and 5  to the back of the answer sheet.

(There will ultimately be 5 problems)


1. Consider the specific wave equation
                 (d/dt)^2 u   = 16 (d/dx)^2  u.
Find a specific constant c so that u(x,t) = f(x - ct) + g(x + ct) solves the
given wave equation for any function f and g.











2.(See 10.1) Find a solution to the boundary value problem

          u"(x)  + A^2  u(x)  = 4 sin x  + 12 sin 5x  + 3 sin 9x

with u(0) = u(pi) = 0, where A   is an arbitrary constant.  For which
numbers A does the method fail? 







3.Find the general solution to (d/dx)^2 u  + (d/dy)^2 u = 0 for
the variable x to be an angle (the solution is assumed periodic of
period 2pi in  the x variable).  Use separation of variables and
leave your answer  as an infinite sum with coefficents left arbitrary.  

              






4.Practice with the maple program on Fourier series set up for this course.
You get to it by typing xmaple step.mws. Run through the example which is
set up to be done with sine series?

What is the function set up?

What is the period?

What are the first 5 Sine  series  coefficents?

What is the value of the maximum of the 5-step approximation?

What is the value of the maximum of the 30-step approximation?

5.  Repeat with the new function piecewise(x<.5,x,x>.5, 1-x).
(It has the same period).

What are the first 5 sine series coefficients?

What is the value of the maximum to the solution to the heat equation
at t = .4?

Which function is better approximated by the sine series?

Note...the questions are somewhat fake, but you should try the program!