MATH 2280-002
Exam 1 Extra Credit (#6)
Math 2280 Section 002 [SPRING 2013]
Instructions: This is extra credit. It’s optional and won’t hurt your grade whether or not you
attempt it. The points you score on this assignment will be used to augment your score on question
#6 of Exam 1. Regardless of how well you perform on this assignment, you will be allowed to score
at most 7 out of 10 points on #6. In other words, if you scored 4/10 on #6, you can earn at most
3 points on this assignment. I have indicated how much each problem is worth so that, if you do
not need many points, you will not need to do as much work. You can attempt as many problems
as you like.
This assignment is due in class on Friday, March 22. You will receive either full or no credit on each
problem. I will be grading more harshly than usual because this is extra credit and I’m anticipating
a lot of questions to grade. If you don’t show enough work, you will receive no points. You need to
convince me that you didn’t just plug the differential equation into Maple or copy another student’s
work. If your work is too messy for me to follow easily, you will receive no points. Be sure to clearly
indicate your final answer. Don’t included unrelated scratch work.
For problems 1-7, find the general solution to the given DE using the separation of variables method.
Find the critical points, and use a phase diagram to determine the stability of each critical point.
1.
2.
3.
4.
5.
6.
7.
( 31
( 13
( 31
( 31
( 13
( 31
( 13
pt)
pt)
pt)
pt)
pt)
pt)
pt)
y′
y′
y′
y′
y′
y′
y′
=y+1
= y(y − 2)
= (1 − y)4
= y2 + 1
= y(y − 5) + 6
= y(y 2 − 9)
= (y + 1)2 (y − 1)
For problems 8-10, find the particular solution to the given IVP using the separation of variables
method. Find the critical points, and use a phase diagram to determine the stability of each critical
point.
8. ( 13 pt)
9. ( 31 pt)
10. ( 13 pt)
dy
= 2y(y − 4), y(0) = 1
dx
dy
= 2y(4 − y), y(0) = 1
dx
dy
= cos2 (y), y( π4 ) = 3
dx
11. ( 31 pt) For problems #1 − 3 above, sketch (by hand) the typical solution curves to the given
DE. Clearly label every critical point. (Each problem should have a separate graph.)
12. ( 13 pt) For problems #5 − 7 above, sketch (by hand) the typical solution curves to the given
DE. Clearly label every critical point. (Each problem should have a separate graph.)
13. ( 23 pt) Suppose a population P is modeled by the equation
dP
= 20P 2 − 1400P.
dt
1
MATH 2280-002
(a)
(b)
(c)
(d)
(e)
Exam 1 Extra Credit (#6)
Find the general solution to the differential equation using separation of variables.
What are the critical values of this differential equation?
Draw a phase diagram and classify each critical value as stable or unstable.
What happens to the population over time if P (0) = 10? Why?
What happens to the population over time if P (0) = 100? Why?
14. ( 13 pt) You want to dissolve the chemical substance Unobtainium in methanol. The number
x(t) of grams of Unobtainium in a solution after t seconds satisfies the differential equation
dx
= 0.4x − 0.001x2 .
dt
(a) What is the maximum amount of Unobtainium that will ever dissolve in the methanol?
(b) If x = 50 when t = 0, how long will it take for an additional 50 g of Unobtainium to
dissolve?
15. ( 31 pt) A population P (t) of small rodents has birth rate β = 0.002P (births per month per
rodent) and a constant death rate δ. If P (0) = 100 and P ′ (0) = 4 how long in months will it
take the population to double to 200 rodents?
16. ( 13 pt) Suppose that at a time t = 0, half of a population of 50,000 people are infected with
the LQP-79 virus and that the number of infected people increases at a rate of 1000 persons
per day. How long until 75% of the population is infected?
17. ( 23 pt) Consider the differential equation
dx
= 0.1x2 − 35x.
dt
Solve this DE using separation of variables.
(a) Suppose x(0) = 500. In this case, there is a vertical asymptote. Find where the asymptote
occurs and describe the behavior around this asymptote.
(b) Suppose x(0) = 50. In this case, there is a horizontal asymptote. What is the horizontal
asymptote? Justify your answer.
(c) In both of these two cases, explain what the phase diagram tells you about the behavior of
the graph.
18. ( 23 pt) Consider an animal population P (t) with constant death rate δ = 0.03 (deaths per
animal per month) and with a birth rate β proportional to P . Solve the corresponding DE
using separation of variables.
(a) Explain why, if P (0) = 600 and P ′ (0) = 9, doomsday occurs. When does doomsday occur?
(b) Explain why, if P (0) = 600 and P ′ (0) = −9, extinction occurs. When does extinction
occur? (Interpret less than one animal as no animal.)
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