Math 5110/6830
Instructor: Alla Borisyuk
Homework 2.1
1. Find and plot solution of the following equation by iteration (up to n = 20) starting with x
0
= 0 .
5.
x n
+1
= 3 .
7 x n
(1
− x n
)
2. (EK 1.1, 1.2) Solve the following difference equations subject to the specified initial conditions and sketch the solution (by hand!). Describe the solutions (increases, decreases, oscillations) a) x n
−
5 x n
−
1
+ 6 x n
−
2
= 0; x
0
= 2 , x
1
= 5.
b) b n
+2
+ 0 .
3 b n
+1 −
0 .
1 b n
= 0; b
1
= 1 , b
2
= 2
3. (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: x n and y n a) x n
+1 y n
+1
= 3 x n
= x n
+ 2 y
+ 4 y n n (1) with x
0
= 1, y
0
= 3.
b) x y n
+1 n
+1
=
=
− x n y n
3
+ 3 y n (2) with x
0
= 2, y
0
= 3.
4. (EK 1.14) In 1202 Fibonacci (Leonardo of Pisa) posed and solved the following problem
(perhaps, the first 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 define:
R 0 n
= number of newborn pairs in generation n .
a) Show that
R
0 n
+1
= R
0 n
+ R
0 n
−
1
.
Can you explain in words why this result makes sense.
b) Starting with a single newborn pair in the first generation ( R 0
0
= 1), how many pairs will there be after n generations? (Find a formula for the solution) c) (extra credit) x
1
The relation that we derived in a): x n
+2
= x n
+1
+ x n with x
0
= 0,
= 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?
1