MTH 370, Fall 2009
Solutions to Homework 1
1. (a) cn+1 = (1 p)kcn
(b) cn = (1 p)k )n c0
(c) cn = c0 p = 1 1/k
2. (a) cn+1 = kcn h
(b) cn = k n c0 h
n1
i=0
k i = k n c0 h kk11
n
(c) cn = c0 k n c0 h kk11 = c0 h = (k 1)c0
n
3. (a) c = 0, and c = ln(r )
(b)
MTH 370, Fall 2009
Solutions to Homework 2
1. Does the dierence equation
cn+1 = cn (3.5 35cn )
have a solution of period 2? If so, is it stable?
Solution:
Factor 3.5 out of the RHS of cn+1 = cn (3.5 35cn ) to get
cn+1 = 3.5cn (1 10cn ).
Multiply both side
MTH 370, Fall 2009
Solutions to Homework 3
1. Let
A=
11
,
11
B=
12
34
(a) Show that AB does not equal BA.
(b) Find a matrix C = A that commutes with A, that is, AC = CA.
Solution:
(a)
46
3
=
46
7
AB =
3
= BA
7
(b) Any multiple C = mI of the identity matri
MTH 370, Fall 2009
Solutions to Homework 4
Instructions: Do these calculations by hand (you may use a computer or calculator for simple arithmetic
and function evaluations) and show your work.
1. Recall from class that the characteristic equation of the m
MTH 370, Fall 2009
Solutions to Homework 5
Instructions: Do these calculations by hand (you may use a computer or calculator for simple arithmetic
and function evaluations) and show your work.
1. Suppose that a population of hosts and parasitoids follow t
MTH 370, Fall 2009
Solutions to Homework 6
Instructions: Do these calculations by hand (you may use a computer or calculator for simple arithmetic
and function evaluations) and show your work.
1. Solve the following rst-order ODEs using either separation
MTH 370, Fall 2009
Solutions to Homework 7
Instructions: Do these calculations by hand (you may use a computer or calculator for simple arithmetic
and function evaluations) and show your work.
1. Solve the following systems of rst-order linear ODEs. In ea
MTH 370, Fall 2009
Solutions to Homework 8
Instructions: Do these calculations by hand (you may use a computer or calculator for simple arithmetic
and function evaluations) and show your work.
1. One unrealistic feature of the the Lotka-Volterra model is
MTH 370, Fall 2009
Solutions to Homework 9
Instructions: Do these calculations by hand (you may use a computer or calculator for simple arithmetic
and function evaluations) and show your work.
1. Write down the mass-action equations for the following chem
MTH 370, Fall 2009
Solutions to Homework 10
Instructions: Do these calculations by hand (you may use a computer or calculator for simple arithmetic
and function evaluations) and show your work.
1. In the real world, trimolecular reactions are rare, althou
MTH 370, Fall 2009
Solutions to Homework 11
Instructions: Do these calculations by hand (you may use a computer or calculator for simple arithmetic
and function evaluations) and show your work.
1. Consider the following reactions:
k1
X A,
k
k
3
2X + Y 3X
MTH 370, Fall 2009
Solutions to Homework 12
Instructions: Do these calculations by hand (you may use a computer or calculator for simple arithmetic
and function evaluations) and show your work.
1. Show that the two-species competition model
dx
x + 12 y
=
MTH 370, Fall 2009
Solutions to Midterm
Name:
Instructions:
1. Print your name in the space provided above.
2. There are two problems. Do these problems by hand (no calculators, computers, etc.) and show your
work on the pages provided (no additional scra
MTH 370, Fall 2009
Midterm
Name:
Instructions:
1. Print your name in the space provided above.
2. There are two problems. Do these problems by hand (no calculators, computers, etc.) and show your
work on the pages provided (no additional scratch paper).
3