PHZ 3113 Fall 2011 — Practice Exam 3
DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO
Instructions: Attempt all four questions. The maximum possible credit for each question is shown in square brackets. Please try to write your solution neatly and legibly.
You will receive credit only for knowledge and understanding that you demonstrate in
your written solutions. It is in your best interest to write down something relevant for every
question, even if you can’t provide a complete answer. To maximize your score, you should
briefly explain your reasoning and show all working. Give all final algebraic answers in terms
of variables defined in the problem.
During this exam, you may use up to three formula sheets. You are not permitted (a)
to consult any other books, notes, or papers, (b) to use any electronic device, or (c) to
communicate with anyone other than the proctor. In accordance with the UF Honor Code,
by turning in this exam to be graded, you affirm the following pledge: On my honor, I have
neither given nor received unauthorized aid in doing this assignment.
Print your name where indicated below, and sign to confirm that you have read and
understood these instructions. Please do not write anything else below the line.
Name (printed):
Signature:
Question
1
2
3
4
Total
Score
In answering the questions below, you may find useful
Z
Z
1
x
x
1
x cos kxdx = 2 cos kx + sin kx,
x sin kxdx = 2 sin kx − cos kx.
k
k
k
k
1. [20 points] The two parts of this question are unrelated.
(a) A mechanical system is described by the equations of motion
d2 x1
d2 x2
2
=
−Ω
(x
+
αx
),
= αΩ2 x1 ,
1
2
2
2
dt
dt
1
where 0 < α < 2 . Find the normal-mode angular frequencies of the system.
(b) Find the determinant and the inverse of the matrix


−1 3
.
A=
−1 1
2. [20 points] Consider the matrix

B=

9 4
4 3
.
(a) Find the eigenvalues λ1 and λ2 of B, defined such that λ1 < λ2 .
(b) Find a normalized eigenvector vj corresponding to each λj .
 
1
(c) Suppose that we want to express the vector   in the form c1 v1 + c2 v2 . Find
1
c1 and c2 .
3. [30 points] Use the Frobenius method find a solution of the differential equation
d2 y
+y =0
dx2
with a0 6= 0.
x
of the form y =
P∞
n=0
an xn+s
(a) Find and solve the indicial equation for s.
(b) Show that one solution of the indicial equation does not lead to a solution of the
form given above.
(c) For the other solution of the indicial equation, find the recurrence relation between
an and an+1 .
4. [30 points] Express the function

 0 −π < x < 0
f (x) =
 x 0<x<π
as a sine and cosine Fourier series.
2