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
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:
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
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
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
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
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 To Test 2
( In-Class Portion )
Math 355, Section 1
Professor J. M. Cushing
1. (20 points) Identify the phase plane portrait for the systems x = Ax with coecient matrices below.
stable node
unstable node
saddle
center
stable spiral
unstable spiral
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
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
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 355-001
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 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 #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 355-002
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
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
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
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
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: [email protected]
webpage: http:/math.arizona.edu/~cushing/
V
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: [email protected]
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 #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 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 #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 #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
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
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 +