Homework 11   427K-H   Due Thursday April 30



1. Find the fundamental period of the functions

a) sin(x)cos(3x),   b) sin^2(48 \pi x),   c) 1 / (1 + tan^2(8x)),

d)  f(x) = x - n(x) where n(x) is the   greatest integer less than x,

c) sin^2(25 x).

2. Which of the functions below are even, odd or neither even nor odd.

a) sin(x)cos(x),  b)  x^2 + x^4,  c) x^12 + x,  d)  |x| - x^2

e)  sin^4(5x).
  

3.  Separate variables in the wave equation 

            (d/dt)^2 u  -  (d/dx)^2 u =0.

 Now use the fact that x is a variable in a circle of radius R whose
circumference is 2 \pi R.    So the boundary conditions   you want to use
can be stated  as    u(0) = u(2\pi R) and    d/dx u (0) = d/dx u (2 \pi R).
These boundary  conditions can also be  stated as the condition that
u(x) is periodic of  period 2 \pi R. Find the general solution.

4. In the equations below, some can be handled by separation of variables
and some cannot.  Separate variables in the equations in which it is
possible, and indicate when it is impossible.

a)d/dt u  - (d/dx)^2 u  - 1/x d/dx u = 0.

b) (d/dz)^2 u  + x (d/dx) u  + x^2 (d/dx)^2 u = 0.

c) d/dx + (x + y) d/dy = 0.

d) (d/dx)(d/dt) u  +  u = 0.


5.Separate variables in the equation  (d/dx)(d/dt)u = 0. Show that
any function f(x,t) = U(x) + V(t)  is a solution of the equation. 


Last extra credit   Due Friday May 1


page 601 #36.

page 602 #37