Math*2270, Fall 2012
Due Wednesday, October 31 in class
▯ Ensure that your name and ID Number are clearly printed on the front page.
▯ You are encouraged to work with your friends to complete the assignment, but please write
up your ▯nal solutions on your own.
▯ Submit your assignment in class on or before the due date.
▯ This assignment carries a weight of 4% of your grade. Late submissions will be given a zero.
1. Consider the second-order DE given by
+ ! x = 0;
where ! > 0 is a constant. This DE is used to model the simple harmonic oscillator,
where ! is the natural frequency of the oscillation. In this example, let’s assume
that the equation models a pendulum with zero friction, with x(t) being the
horizontal position of the bottom of the pendulum at time t.
a) Find the general solution to the DE. Show that if the initial position of the
pendulum is zero and it has no initial velocity, that the pendulum stays in the same
place for all time. Show also that if either the initial position or velocity is nonzero,
that the pendulum moves in a periodic fashion for all time. (Note: Periodicity of x
implies that for any time t, x(t) = x(t + T), where T is a constant.)
b) Now, consider a modi▯ed version of this DE that takes into consideration friction.
Friction opposes motion, and so the model includes a term involving velocity x _:
+ 2▯x_ + ! x = 0;
where ▯ >