Math 5110/6830 Instructor: Alla Borisyuk Homework 2.1 Due: September 8

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Math 5110/6830
Instructor: Alla Borisyuk
Homework 2.1
Due: September 8
1. (EK 1.1, 1.2) Solve the following dierence equations subject to the specied values and sketch the solution (by hand!). Describe the solutions
(increases, decreases, oscillations, characteristic time scale)
a) n 5 n 1 + 6 n 2 = 0; 0 = 2 1 = 5.
b) n + n 2 = 0; 1 = 3 2 = 5.
c) n+2 3 n+1 + 2 n = 0; 0 10 1 = 20
2. (EK 1.6, 1.7) In each of the following systems determine the eigenvalues
(from reduction to a single equation, or from a matrix). Find the solution
of the system and sketch the solution. You will have two components for
each solution: n and n
x
x
x
x
x
x
x
x
x
x
;x
x
x
x
;x
:
y
a)
n+1
n+1
= 3 n+2 n
= n+4 n
(1)
n+1
n+1
=
n+3
= y3n
(2)
x
y
with 0 = 1, 0 = 3.
x
;x
x
y
x
y
y
b)
x
y
x
y
n
with 0 = 2, 0 = 3.
3. (EK 1.14) In 1202 Fibonacci (Leonardo of Pisa) posed and solved the
following problem (perhaps, the rst mathematical idealization of a biological phenomenon). Suppose that every pair of rabbits can reproduce
only twice, when they are one and two months old, and that every time
they produce exactly one new pair of rabbits (one male and one female).
Assume that all rabbits survive. To solve the problem, let us dene:
0
n = number of newborn pairs in generation ,
1
n = number of one-month-old pairs in generation ,
2
n = number of two-month-old pairs in generation .
a) Show that
0
0
0
n+1 = n + n 1
Can you explain in words why this result makes sense.
b) Starting with a single newborn pair in the rst generation ( 00 = 1),
how many pairs will be there after generations?
c) (extra credit) The relation that we derived in a): n+2 = n+1 + n
with 0 = 0, 1 = 1 generates a sequence of numbers, known as Fibonacci
numbers. List a few examples of where these numbers are found (in nature,
art, or geometry or elsewhere). You can use anything (yes, google also) to
answer this. Why do you think these numbers are so common?
x
y
R
n
R
n
R
n
R
R
R
:
R
n
x
x
x
1
x
x
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