MATH 355-001
Fall 2012
ANSWERS HAND-IN HOMEWORK #4 (due Thursday, September 27)
Show your work and justify answers.
Chapter 3.1.1 (pages 133-134), exercise 1.11: How many equilibria does the equation x =
x4 xeax where a > 0 is a positive constant?
ANSWER:
TEST 1
Take-Home Part
Math 355
Professor J. M. Cushing
DUE DATE: In class on Tuesday, 28 October, 2014. Total points = 100
1. (50 points) In class we derived the initial value problem
0 = 00231 001
0.01d
ce
(0) = 1
Bacterialkillrate
for an infection of 1
MATH 355-001
ANSWERS HAND-IN HOMEWORK #2
(due Thursday, September 12)
PART 1
Pages 61-72, exercise 241. Find a formula for the general solution of the following
linear homogeneous equation:
1
x=
x.
t t2
1
ANSWER. From p (t) = tt2 we calculate (using, for
Answers Test 1 - Take Home
Math 355, Section 2
1 March 2012
Professor J. M. Cushing
A manufactured item is given an initial set price of x0 . However, as supply and demand for the
item changes with time t, so does the price x = x(t). The supply and demand
ANSWERS: In-class Test #1
Math 355
10 March 2015
1. (10 points) Consider the initial value problem
0 = (sin( + )13 (0) = 0
(a) For what values of 0 does the Fundamental Existence and Uniqueness Theorem apply?
Justify your answer.
ANSWER: The polynomial a
TEST 1 (Math 355-002)
Test #1, Take-Home Part
Professor J. M. Cushing
5 March 2013
1. (50 points) A basic model for population growth is the balance law x = xdx or x = ( d) x
where is the per capita birth rate and d is the per capita death rate. In this p
ANSWERS TEST 1
( Take-Home Portion - Due October 25 )
Math 355, Section 1
18 October 201
(1) (40 points) (Chapter 3.4, page 190: exercise 4.27). Consider an object falling under the inuence
of gravity. If we assume the acceleration due to gravity is a con
MATH 355-002
Spring 2013
ANSWERS HAND-IN HOMEWORK #9
(due Thursday, April 11)
Chapter 5.4 (pages 337-340), exercise 4.2: Identify the
type of phase portrait and sketch it for the system:
x
y
y
= 2x + y
= x 2y.
2
ANSWER: The coecient matrix
A=
x
-4
2 1
1 2
MATH 355-002
Spring 2013
ANSWERS HAND-IN HOMEWORK #8
(due Thursday, April 4)
Chapter 5.5 (pages 341-345), exercise 5.10: Find a fundamental solution matrix and
the general solution for the system
x0
y0
= 3x + 2y
= 4x y
ANSWER: The coecient matrix
A=
3
2
4
MATH 355-001
Fall 2012
ANSWERS HAND-IN HOMEWORK #5 (due Thursday, October 4)
Show your work and justify answers.
Chapter 3.1.3 (pages 145-146), exercise: 1.52: Find the linearization, at each of its equilibria,
of the equation
x2
x = 2x 4
1 + x2
ANSWER: W
MATH 355-001
Fall 2012
ANSWERS HAND-IN HOMEWORK #6 (due Thursday, October 11)
Show your work and justify answers.
Chapter 3.2 (pages 168-170), exercise 2.2: Find a formula for the general solution of
the equation x0 = 1 x2 .
ANSWER: This is an autonomous
MATH 355-001
Fall 2012
ANSWERS HAND-IN HOMEWORK #1 (due Thursday, September 6)
Chapter 1 (p 32): Exercise 1.20. For what values of the constant a can the Fundamental
Existence and Uniqueness Theorem 1.1 be applied to the initial value problem below? Expla
ANSWERS TEST 1
In-Class Part
Math 355-002
28 February 2012
Professor J. M. Cushing
1. (20 points) Consider the dierential equation
x = (1 x)1/3 (1 t)1/3 .
(a) Explain why the Fundamental Existence & Uniqueness Theorem does or does not apply
to the initial
MATH 355-001
Fall 2012
ANSWERS HAND-IN HOMEWORK #2 (due Thursday, September 13)
Chapter 2.1 (pages 76-79), Exercise 1.18: On what interval a < t < b does the solution of
the following initial value problem exist? (Do not nd formulas for solutions. Instead
Answers To Test 1
In-class Part
Math 355-001
8 October 2013
Professor J. M. Cushing
Total possible points = 100.
1. (15 points) Consider the equation:
x0 = x3
2 x2
.
1 + x2
(a) Find all equilibria.
xe = 0,
ANSWER:
2, 2
2 x2
= 0 = x = 0 or 2 x2 = 0.
1 + x2
MATH 355-002
Spring 2013
ANSWERS HAND-IN HOMEWORK #6
(due Tuesday, February 26)
Chapter 3.4.2 (pages 189-190), exercise 4.18: Find the rst order perturbation
expansion for the solution of the initial value problem
x = x + e2t ,
x (0) = 7
ANSWER: A substit
Analysis of Ordinary Dierential Equations
Math 355
J. M. Cushing
Department of Mathematics
Interdisciplinary Program on Applied Mathematics
University of Arizona
Tucson, AZ, 85721
email: cushing@math.arizona.edu
webpage: http:/math.arizona.edu/~cushing/
S
1. Homogeneous Linear Systems
Linear combinations of solutions are solutions.
Two solutions are dependent
det
x1 (t )
x2 (t )
y1 (t )
y2 ( t )
0 for at least one t
The general solution is the span of any two independent solutions
The exist two indepen
MATH 355-001
Fall 2012
ANSWERS HAND-IN HOMEWORK #10 (due Thursday, November 20)
Chapter 7.3 (page 443), exercise 3.8 for 3.2: Apply Theorem 3.1 and/or 3.2, if possible,
to determine the stability of each equilibrium of the system
x0
y0
= x (2x y )
= x2 +
MATH 355-001
Fall 2012
ANSWERS HAND-IN HOMEWORK #9 (due Thursday, November 15)
4
Chapter 5.4 (pages 337), exercise 4.2, parts (a) & (b).
Identify and sketch the phase portrait of the system
y
x0
y0
2
x
4
2
2
2
4
4
= 2x + y
= x 2y.
ANSWER: The eigenvalues
MATH 355-002
Spring 2013
ANSWERS HAND-IN HOMEWORK #2 (due Thursday, January 31)
Chapter 2.1 (pages 74-79): exercise 1.20. On what interval a < t < b do the solutions
of the following initial value problems exists? (Use Corollary 1.1.)
x=
1
1
x+
,
sin 2t
c
MATH 355-001
Fall 2012
ANSWERS HAND-IN HOMEWORK #7 (due Thursday, October 25)
Chapter 4.1 (pages 239-240), exercise: 1.15: Determine those initial conditions t0 , x0
and y0 for which the Fundamental Existence & Uniqueness theorem applies to the
system
x
y
ANSWERS To Test 2
( In-Class Portion )
Math 355, Section 1
Professor J. M. Cushing
25 April 2013
1. (15 points) Identify the type of phase portraits for the system x0 = Ax with each of the
coecient matrices below. Show your work
13
(a) A =
42
ANSWER: The
ANSWERS HAND-IN HOMEWORK #3
Spring Semester 2015
( Due Thursday, February 12 )
1. Page 64, exercise 304: Use the Method of Undetermined Coecients to (a) formulate
a guess for a particular solution () of the equation below. (b) Use your guess to nd a
parti
ANSWERS HAND-IN HOMEWORK #1
(due Thursday, September 5)
Page 30, exercise 128. Does the Fundamental Existence and Uniqueness Theorem 1
apply to the initial value problem below?
x=
1
,
tx
x (1) = 2
Explain your answer. What do you conclude from the theorem
MATH 355-001
ANSWERS HAND-IN HOMEWORK #4
(due Thursday, September 26)
Part 1
1. Pages 105-107, excercise 432: For the equation below nd all equilibria. Sketch a
graph of f (x) and use it to draw the phase line portrait. Identify the type of each
equilibri
MATH 355-001
ANSWERS HAND-IN HOMEWORK #5
(due Thursday, October 3)
1. Page 108, exercise 497.
diagram for (excercise 497)
Draw a bifurcation
x
1
x = p x3 ,
stable
< p < +.
p
Include in the diagram the type of each equilibrium.
2
1
1
2
stable
1
ANSWER. Fr
MATH 355-001
ANSWERS HAND-IN HOMEWORK #3
(due Thursday, September 19)
1. Pages 71-72, excercise 399: Without nding a formula for the solution
of the equation determine the asymptotic dynamics of each of the following
autonomous equations. Draw the phase l
MATH 355-001
Fall 2012
ANSWERS HAND-IN HOMEWORK #8 (due Thursday, November 8)
Chapter 5.5 (pages 345-347), exercise 5.2: Find a fundamental solution matrix and the
general solution for the system
x
y
= 2x + y
= x 2y
ANSWER: Eigenvalues and eigenvectors of
MATH 355-001
Fall 2012
ANSWERS HAND-IN HOMEWORK #3 (due Thursday, September 20)
Chapter 2.3 (pages 91-92), exercise 3.12: (a) Construct the appropriate Undetermined Coecients guess for a particular solution xp and (b) use the guess to nd a particular solu