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Math - College, section 1, Fall 2019
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MacCluer et al - Differential Equations 1/e (Homework)

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 1 / 15

Due : Sunday, January 27, 2030 00:00 EST

Last Saved : n/a Saving... ()

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Instructions

Differential Equations: Techniques, Theory, and Applications is designed for a modern first course in differential equations either one or two semesters in length. The organization of the book interweaves the three components in the subtitle, with each building on and supporting the others. Techniques include not just computational methods for producing solutions to differential equations, but also qualitative methods for extracting conceptual information about differential equations and the systems modeled by them. Theory is developed as a means of organizing, understanding, and codifying general principles. Applications show the usefulness of the subject as a whole and heighten interest in both solution techniques and theory. Formal proofs are included in cases where they enhance core understanding; otherwise, they are replaced by informal justifications containing key ideas of a proof in a more conversational format. Applications are drawn from a wide variety of fields: those in physical science and engineering are prominent, of course, but models from biology, medicine, ecology, economics, and sports are also featured.

The WebAssign enhancement to this textbook is a fully customizable online solution that empowers students to learn, not just do homework, and includes links to a complete eBook. Insightful tools save time and highlight exactly where students are struggling. Students get an engaging experience, instant feedback, and better outcomes.

Questions 1 and 2 highlight real world applications of differential equations.

Question 3 is a multipart question asking the students to perform various analyses of the differential equations.

Question 4 showcases the grading that accepts any unique set of constant parameters used in student responses.

Question 5 is an example of an exercise with randomized parameters.

Question 6 demonstrates interval grading, which can grade any canonically equivalent interval and enforce proper notation.

Question 7 showcases grading for answers that involve lists.

Quesiton 8 is an example of an exercise that provides formulas or other reference content at the point of use. This demo assignment allows many submissions and allows you to try another version of the same question for practice wherever the problem has randomized values.

Assignment Submission

For this assignment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change the answer.

Assignment Scoring

Your last submission is used for your score.

1. 0/2 points | Previous Answers MCDiffEQ1 2.3.020. My Notes

A tank initially contains 1 gram of blue dye dissolved in 10 liters of water. Pure water will be added to the tank at a certain rate, and the well-mixed solution will be drained from the tank at the same rate. The tank is considered "cleaned" when the dye concentration in the tank is no more than 0.001 gr/liter.

(a)

If the rate at which pure water is added (and the tank is drained) is 3 liters/min, how long (in minutes) does it take until the tank is cleaned?

min

(b)

If we want to clean the tank in 6 minutes, how quickly (in liters/min) must the pure water be poured in?

7.7

liters/min

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2. 0/1 points | Previous Answers MCDiffEQ1 2.5.010. My Notes

Suppose that starting at age 25, you make steady contributions to a retirement account (with initial balance 0). What should your yearly contribution be if you want to have a balance of $815,000 after 40 years? Assume your account will earn 7% interest, compounded continuously. (Round your answer to the nearest dollar.)

3,694

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3. –/5 points MCDiffEQ1 2.1.024. My Notes

Various mathematical models have been proposed to study a sprinter running in a straight line (say, a 100-meter race). The simplest of these uses the differential equation

= F_prop − F_res

where the runner's velocity is v and mass is m. The term F_prop is called the "propulsive force" of the runner, and "F_res" is the resistive force; the latter includes frictional losses within the body of the runner.

(a)

As a simple first assumption, we set F_prop to be a constant (individualized to the specific runner being modeled) and assume F_res is proportional to the runner's velocity v. Upon dividing by the mass m we obtain the differential equation

= γ − σv

for positive constants γ and σ (both specific to the individual runner). Solve this equation for v, assuming that

v(0) = 0.

v =

(b)

Recalling that for an object moving along a straight line,

= v

where

p(t)

is the position of the object at time t and v is the velocity, use your work in (a) to determine the distance the sprinter has run as a function of elapsed time. (Round your answer to three decimal places.)

p(t)

(c)

Using data from the first 8 seconds of Usain Bolt's 100-meter final at the 2008 Beijing Olympics, a race that he ran in an astonishing 9.69 sec, and assuming the model in part (a), estimates for Bolt's values of γ and σ can be given as

γ ≈ 8.5269 m/sec²,

σ ≈ 0.6964/sec.

Using these values in your distance function from (b), determine his predicted position (in m) at his finishing time of

t = 9.69;

your answer should be greater than the actual position of 100 m at

t = 9.69.

This leads to an interesting feature of this particular race, which will be explored in the next part of this problem.

(d)

(CAS) Bolt's time of 9.69 sec in the 100-meter final of the 2008 Beijing Olympics was a world record, but what was more remarkable is that about 8 seconds into the race, when about 20 meters remained, Bolt appears to begin to celebrate his victory, causing a significant drop in his speed for the last 20 meters or so. See the following table, which presents "100-meter splits" data often used to analyze this race.

Usain Bolt Beijing Olympics
100 Meter Split Times
Segment	Segment Time	Cumulative
0-10 m	1.85	1.85
10-20 m	1.02	2.87
20-30 m	0.91	3.78
30-40 m	0.87	4.65
40-50 m	0.85	5.50
50-60 m	0.82	6.32
60-70 m	0.82	7.14
70-80 m	0.82	7.96
80-90 m	0.83	8.79
90-100 m	0.90	9.69

This drop in Bolt's speed led many commentators to speculate that he could have had a significantly faster time if he had not begun his celebration until the race was actually finished. His coach, for example, believed that his time could have been 9.52 sec. Based on your work in (b) and (c), you are in a position to predict what Bolt's finishing time might have been, had he not begun celebrating at around the 80-meter mark.

Let

d(t)

be the distance function you derived in (c) based on your work from (b). This function represents Bolt's predicted position (distance from the start line) at a time t under the model of (a) based on data from the first 8 seconds of his run. The predicted distance given by

d(t),

as shown in the following figure, fits the data from the table above quite well—especially over the first 8 seconds.

A plot of a function which represents Bolt's predicted position at a time t. The curve enters the window at the origin, connecting points on the plot. The following points are as follows:

(0, 0),

(1.85, 10),

(2.87, 20),

(3.78, 30),

(4.65, 40),

(5.5, 50),

(6.32, 60),

(7.14, 70),

(7.96, 80),

(8.79, 90),

(9.69, 100)

Using a computer algebra system, find the positive root of

d(t) = 100

to two decimal places. (You might use the system's ability to solve equations numerically or its graphing capability.) The time you have found represents Bolt's predicted finishing time according to the model of (a), based on data from his run up to the 7.96 second mark. (Enter your answer in seconds.)

(e)

Compare your prediction of (d) to those of a group of physicists from the University of Oslo who obtained detailed camera footage from the race and projected Bolt's potential time under two scenarios: for one their projected time is

9.61 ± 0.04

seconds and for the other it's

9.55 ± 0.04.

Remark: In 2009, at the world championships in Berlin, Bolt ran the 100-meter race in 9.58 sec.

Our value in (d) does not agree at all their first prediction (off by more than 20%), but is quite close to the second projection. Our value in (d) agrees very well with their first prediction, and is quite close to the second projection. Our value in (d) does not agree at all their first prediction (off by more than 20%), and is also about 20% from the second projection. Our value in (d) agrees very well with their first prediction, but is more than 20% from the second projection.

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4. 0/1 points | Previous Answers MCDiffEQ1 4.2.006. My Notes

Record the general solution (assuming t is the independent variable).

y″ − 5y′ + 6y = 0

y=C9e4t+C2e3t

$webMathematica generated answer key$

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5. 1/1 points | Previous Answers MCDiffEQ1 4.7.004. My Notes

Give the general solution.

y‴ + 10y″ + 25y′ = 0

y(t) =

15C1e−5t−125C2e−5t−15C2e−5tt+C3

$webMathematica generated answer key$

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6. –/2 points MCDiffEQ1 2.1.014. My Notes

Solve the initial value problem, and give the interval of validity of your solution.

s − 3

= 2, r(2) = 1

r =

Give the interval of validity of your solution. (Enter your answer using interval notation. If you need to use

or –

, enter INFINITY or –INFINITY, respectively.)

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7. 0/2 points | Previous Answers MCDiffEQ1 5.1.016. My Notes

Suppose that the equation

y″ + p(t)y′ + q(t)y = g(t)

has solutions

f₁(t) = t,

f₂(t) = t²,

and

f₃(t) = t³

on an interval I on which p, q, and g are continuous.

(a)

Find a fundamental set of solutions for the associated homogeneous equation

y″ + p(t)y′ + q(t)y = 0.

(Enter your answers as a comma-separated list.)

y =

t2−t, t3−t2

$webMathematica generated answer key$

(b)

Find the solution to the original equation

y″ + p(t)y′ + q(t)y = g(t).

that satisfies the initial conditions

y(t) = 2,

y′(2) = 3.

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8. –/1 points MCDiffEQ1 6.3.006. My Notes

Use the table below to compute the Laplace transform of the given function.

Some Laplace Transforms▲

Some Laplace Transforms

f(t)

F(s) = ℒ[f(t)]

e^at

s − a

s²

tⁿ

s^{n + 1}

n = 1, 2, 3, …

tⁿe^at

(s − a)^{n + 1}

n = 1, 2, 3, …

sin(βt)

s² + β²

cos(βt)

s² + β²

t sin(βt)

2βs

(s² + β²)²

t cos(βt)

s² − β²

(s² + β²)²

e^at sin(βt)

(s − a)² + β²

e^at cos(βt)

s − a

(s − a)² + β²

sinh(βt)

s² − β²

cosh(βt)

s² − β²

u_a(t)

e^−as

sin²(5t)

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MacCluer et al - Differential Equations 1/e (Homework)

Current Score : 1 / 15

Due : Sunday, January 27, 2030 00:00 EST

Last Saved : n/a Saving... ()

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