Differential Equations: An Introduction to Modern Methods and Applications
3rd Edition
ISBN: 9781118531778
Author: James R. Brannan, William E. Boyce
Publisher: WILEY
expand_more
expand_more
format_list_bulleted
Videos
Textbook Question
Chapter 7.2, Problem 1P
In each of Problems
(a) Determine all critical points of the given system of equations.
(b) Find the corresponding linear system near each critical point.
(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?
(d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
(e) Draw a sketch of, or describe in words, the basin of attraction of each asymptotically stable critical point.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
(a) Test the hypothesis.
Consider the hypothesis test Ho
=
:
against H₁o < 02. Suppose that the sample sizes aren₁ =
7 and n₂
= 13 and that
$²
= 22.4 and $22
= 28.2. Use α = 0.05.
Ho
is not
✓ rejected.
9-9
IV
(b) Find a 95% confidence interval on of 102. Round your answer to two decimal places (e.g. 98.76).
Let us suppose we have some article reported on a study of potential sources of injury to equine veterinarians conducted at a
university veterinary hospital. Forces on the hand were measured for several common activities that veterinarians engage in when
examining or treating horses. We will consider the forces on the hands for two tasks, lifting and using ultrasound. Assume that both
sample sizes are 6, the sample mean force for lifting was 6.2 pounds with standard deviation 1.5 pounds, and the sample mean force
for using ultrasound was 6.4 pounds with standard deviation 0.3 pounds. Assume that the standard deviations are known.
Suppose that you wanted to detect a true difference in mean force of 0.25 pounds on the hands for these two activities. Under the null
hypothesis, 40 = 0. What level of type II error would you recommend here?
Round your answer to four decimal places (e.g. 98.7654). Use a
= 0.05.
β
= i
What sample size would be required?
Assume the sample sizes are to be equal.…
=
Consider the hypothesis test Ho: μ₁ = μ₂ against H₁ μ₁ μ2. Suppose that sample sizes are n₁ =
15 and n₂ =
15, that x1 = 4.7
and X2 = 7.8 and that s² = 4 and s² = 6.26. Assume that o and that the data are drawn from normal distributions. Use
απ 0.05.
(a) Test the hypothesis and find the P-value.
(b) What is the power of the test in part (a) for a true difference in means of 3?
(c) Assuming equal sample sizes, what sample size should be used to obtain ẞ = 0.05 if the true difference in means is - 2? Assume
that α = 0.05.
(a) The null hypothesis is
98.7654).
rejected. The P-value is 0.0008
(b) The power is 0.94
. Round your answer to four decimal places (e.g.
Round your answer to two decimal places (e.g. 98.76).
(c) n₁ = n2 =
1
. Round your answer to the nearest integer.
Chapter 7 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problemsthrough:
Find...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problemsthrough:
Find...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problemsthrough:
Find...Ch. 7.1 - For each of the systems in Problems 1 through 18:...
Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problemsthrough:
Find...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problemsthrough:
Find...Ch. 7.1 -
Consider the equations of motion of an undamped...Ch. 7.1 - The motion of a certain undamped pendulum is...Ch. 7.1 - Consider the pendulum equations dxdt=y,dydt=6sinx....Ch. 7.1 - Prob. 22PCh. 7.1 - Given that x=(t),y=(t) is a solution of the...Ch. 7.1 - Prove that, for the system...Ch. 7.1 - Prove that if a trajectory starts at a noncritical...Ch. 7.1 - Assuming that the trajectory corresponding to a...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - Consider the autonomous system dxdt=y,dydt=x+2x3....Ch. 7.2 - Consider the autonomous system
...Ch. 7.2 - The equations of motion of a certain nonlinear...Ch. 7.2 - Theorem 7.2.2 provides no information about the...Ch. 7.2 - In this problem, we show how small changes in the...Ch. 7.2 - In this problem, we show how small changes in the...Ch. 7.2 - A generalization of the damped pendulum equation...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Show that (1X+2Y)24(1212)XY=(1X2Y)2+412XY. Hence...Ch. 7.3 - Consider the system (2) in the text, and assume...Ch. 7.3 - Consider the system (3) in Example 1 of the text....Ch. 7.3 - The system x=yy=yx(x0.15)(x3) Results from an...Ch. 7.3 - Bifurcation points. Consider the system...Ch. 7.3 - Bifurcation points. Consider the system
Where is...Ch. 7.3 - Bifurcation points. Consider the system
Where is...Ch. 7.3 - Bifurcation points. Consider the system
Where is...Ch. 7.3 - In each of Problem 15 and 16: a) Find the critical...Ch. 7.3 - In each of Problem 15 and 16:
Find the critical...Ch. 7.3 - Suppose that a certain pair of competing species...Ch. 7.4 - Each of Problems 1 through 5 can be interpreted as...Ch. 7.4 - Each of Problems 1 through 5 can be interpreted as...Ch. 7.4 - Each of Problems 1 through 5 can be interpreted as...Ch. 7.4 - Each of Problems 1 through 5 can be interpreted as...Ch. 7.4 - Each of Problems 1 through 5 can be interpreted as...Ch. 7.4 - In this problem, we examine the phase difference...Ch. 7.4 - a) Find the ratio of the amplitudes of the...Ch. 7.4 -
Find the period of the oscillations of the prey...Ch. 7.4 - Consider the system
Where and are positive...Ch. 7.4 - The average size of the prey and predator...Ch. 7.4 - In Problems 11 and 12, we consider the effect of...Ch. 7.4 - In Problems 11 and 12, we consider the effect of...Ch. 7.4 - In the Lotka-Volterra equations, the interaction...Ch. 7.4 - Harvesting in a Predator-Prey Relationship. In a...Ch. 7.4 - Harvesting in a Predator-Prey Relationship. In a...Ch. 7.4 - Harvesting in a Predator-Prey Relationship. In a...Ch. 7.5 - In each of Problems through , an autonomous...Ch. 7.5 - In each of Problems 1 through 6, an autonomous...Ch. 7.5 - In each of Problems 1 through 6, an autonomous...Ch. 7.5 - In each of Problems 1 through 6, an autonomous...Ch. 7.5 - In each of Problems through , an autonomous...Ch. 7.5 - In each of Problems 1 through 6, an autonomous...Ch. 7.5 - If x=rcos,y=rsin, show that...Ch. 7.5 - (a) Show that the system has periodic solutions...Ch. 7.5 - Determine the periodic solutions, if any, of the...Ch. 7.5 - Using Theorem, show that the linear autonomous...Ch. 7.5 - In each of Problems 11 and 12, show that the given...Ch. 7.5 - In each of Problems and , show that the given...Ch. 7.5 - Prob. 13PCh. 7.5 -
By examining the graphs of vs. in Figures , , ...Ch. 7.5 - The equation u(113u2)u+u=0 Is often called the...Ch. 7.5 - Consider the system of equations...Ch. 7.5 - Consider the van der Pol system x=y,y=x+(1x2)y,...Ch. 7.5 - Problems 18 and 19 extend the consideration of the...Ch. 7.5 - Problems 18 and 19 extend the consideration of the...Ch. 7.5 - There are certain chemical reactions in which the...Ch. 7.5 - The system
Is a special case of the...Ch. 7.6 - Problems through ask you to fill in some of the...Ch. 7.6 - Problems through ask you to fill in some of the...Ch. 7.6 -
Ch. 7.6 - Consider the ellipsoid
.
Calculate
along...Ch. 7.6 - In each of Problems 5 through 7, carry out the...Ch. 7.6 - In each of Problems 5 through 7, carry out the...Ch. 7.6 - In each of Problems 5 through 7, carry out the...Ch. 7.6 - For certain intervals, or windows, the Lorenz...Ch. 7.6 - Now consider values of r slightly larger than...Ch. 7.P1 - Assume that , that is, the total size of the...Ch. 7.P1 - The triangular region in the SI-plane is depicted...Ch. 7.P1 - If epidemics are identified with solution...Ch. 7.P1 - Find an equation of the form satisfied by the...Ch. 7.P1 - In the SIR system (1), describe qualitatively the...Ch. 7.P1 - Vaccinated individual are protected from acquiring...Ch. 7.P1 - Use the equation to reduce the SIRS model (3) to...Ch. 7.P2 - Consider again the system
(i)
Which...Ch. 7.P2 - Consider the system dxdt=x(1xy),dydt=y(0.80.6yx),...Ch. 7.P2 - Consider the system (i) in Problem 1, and assume...Ch. 7.P2 - Aconstant-yield model, applied to species x,...Ch. 7.P3 - a) Show that there are no critical points when...Ch. 7.P3 - a) Let c=1.3. Find the critical points and the...Ch. 7.P3 - The limit cycle found in Problem 2 comes into...Ch. 7.P3 -
Let. Find the critical points and the...Ch. 7.P3 -
Let. Find the critical points and the...
Additional Math Textbook Solutions
Find more solutions based on key concepts
8. Effect of Blinding Among 13,200 submitted abstracts that were blindly evaluated (with authors and institutio...
Elementary Statistics
For a population containing N=902 individual, what code number would you assign for a. the first person on the ...
Basic Business Statistics, Student Value Edition
A system is composed of 5 components, each of which is either working or failed. Consider an experiment that co...
A First Course in Probability (10th Edition)
CHECK POINT I You deposit $1000 in a saving account at a bank that has a rate of 4%. a. Find the amount, A, of ...
Thinking Mathematically (6th Edition)
Fill in each blank so that the resulting statement is true. An equation that expresses a relationship between t...
Algebra and Trigonometry (6th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- Consider the hypothesis test Ho: = 622 against H₁: 6 > 62. Suppose that the sample sizes are n₁ = 20 and n₂ = 8, and that = 4.5; s=2.3. Use a = 0.01. (a) Test the hypothesis. Round your answers to two decimal places (e.g. 98.76). The test statistic is fo = i The critical value is f = Conclusion: i the null hypothesis at a = 0.01. (b) Construct the confidence interval on 02/022 which can be used to test the hypothesis: (Round your answer to two decimal places (e.g. 98.76).) iarrow_forwardConjecture Let x and y be integers. If x is even and y is odd, then xy is even. Try some examples. Does the conjecture seem to be true or false?arrow_forwardFind the general solution to the differential equationarrow_forward
- 3. A room has a large circular table with ten seats, numbered 1 to 10, such that to the right of seat number i is seat number i + 1 for all i ∈ {1, . . . , 9} and to the right of seat 10 is seat 1. We want to assign seats to 10 people, 6 of them only speak Slovene, 1 of them only speaks English, and the remaining 3 speak both Slovene and English, by giving out numbered place cards. In how many ways can we do that so that everyone sits next to at least one person who speaks a common language?arrow_forwardcharity savings Budget for May travel food Peter earned $700 during May. The graph shows how the money was used. What fraction was clothes? O Search Submit clothes leisurearrow_forwardExercise 11.3 A slope field is given for the equation y' = 4y+4. (a) Sketch the particular solution that corresponds to y(0) = −2 (b) Find the constant solution (c) For what initial conditions y(0) is the solution increasing? (d) For what initial conditions y(0) is the solution decreasing? (e) Verify these results using only the differential equation y' = 4y+4.arrow_forward
- SOLVE ONLY FOR (L) (M) AND (O)arrow_forwardAphids are discovered in a pear orchard. The Department of Agriculture has determined that the population of aphids t hours after the orchard has been sprayed is approximated by N(t)=1800−3tln(0.17t)+t where 0<t≤1000. Step 1 of 2: Find N(63). Round to the nearest whole number.arrow_forward1. A telegraph can transmit two different signals: a dot and a dash. We want to encode the 26 letters of the English alphabet and the ten digits 0, 1, 2, . . . , 9 using sequences of these two symbols. What is the smallest integer n such that we can encode all these letters and digits with sequences of length at most n and length at least 1?arrow_forward
- Use the graph of y = f(x) to answer the following. 3- 2 -4 -2 -1 1 2 3 4 -1 2 m -3- + (d) Find all x for which f(x) = -2. If there is more than one value, separate them with commas or write your answer in interval notation, if necessary. Select "None", if applicable. Value(s) of x for which f(x)=-2: | (0,0) (0,0) (0,0) (0,0) 0,0... -00 None (h) Determine the range of f. The range is (0,0) Garrow_forwardFile Preview A gardener has ten different potted plants, and they are spraying the plants with doses of Tertizers. Plants can receive zero or more doses in a session. In the following, we count each possible number of doses the ten plants can receive (the order of spraying in a session does not matter). (a) How many ways are there if there were twelve total doses of a single type of fertilizer? (b) How many ways are there if there are six total doses of a single type of fertilizer, each plant receives no more than one dose? (c) How many ways are there if is was one dose of each of six types of fertilizers? (d) How many ways are there if there are four doses of fertilizer #1 and eight doses of fertilizer #2? (e) How many ways are there if there are four doses of fertilizer #1 and eight doses of fertilizer #2, and each plant receives no more than one dose of fertilizer #1? (f) How many ways are there to do two sessions of spraying, where each plant receives at most two doses total?arrow_forward3. [-/3 Points] DETAILS MY NOTES SCALCET8 7.4.032. ASK YOUR TEACHER PRACTICE ANOTHER Evaluate the integral. X + 4x + 13 Need Help? Read It SUBMIT ANSWER dxarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Lecture 46: Eigenvalues & Eigenvectors; Author: IIT Kharagpur July 2018;https://www.youtube.com/watch?v=h5urBuE4Xhg;License: Standard YouTube License, CC-BY
What is an Eigenvector?; Author: LeiosOS;https://www.youtube.com/watch?v=ue3yoeZvt8E;License: Standard YouTube License, CC-BY