Assignment 11   427K-H   Due Wednesday, April 28 



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

 Now use the fact that x is an  angle variable in a circle, so solutions
must be periodic of period 2 pi in the x variable. 
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. 







.