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MATH 333 Quiz 4: Solutions to Differential Equations and Population Dynamics, Quizzes of Differential Equations

The questions and solutions for quiz 4 of math 333, which covers topics such as solving first order differential equations and population dynamics. The quiz includes finding general solutions to differential equations, modeling real-world processes with differential equations, and analyzing equilibrium solutions in population dynamics.

Typology: Quizzes

Pre 2010

Uploaded on 08/19/2009

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MATH 333 Quiz 4
You may work with other class members on this quiz, but you may not receive
assistance from people not in MATH 333 (Section 002). You must show all of
your work to receive full credit. Do all your work on other sheets of paper and
be sure to staple all the pieces of paper together or YOU WILL GET A ‘ZERO’
ON THE QUIZ. Do not use decimal approximations unless asked to do so. Your
work on this quiz must be handed in by Friday, 3 October 2008 at 1040. GOOD
LUCK!
1) Let t > 0. Find the general solution of
dy
dt +3y
t=cos 2t
t2
2) A 100-gallon tank is half full of pristine water. A salt water solution with
5 pounds of salt per gallon enters the tank at 3 gallons per minute and at the
same time the well-stirred solution leaves the tank at 4 gallons per minute.
a) Write an initial value problem that models this process.
b) Over what time frame does the differential equation model this process?
Explain.
c) Solve the initial value problem.
d) At what time is the amount of salt in the tank a maximum? Give the
exact solution and an appropriate estimate.
3) Consider the population mo del
dR
dt =1
4R1
R
3
1
4RF
dF
dt =
1
2F+1
3RF
(1)
a) Find all equilibrium solutions of (1).
b) Describe completely what happens to species Fif species Ris extinct.
c) Describe completely what happens to species Rif species Fis extinct.
d) Depict in the R-Fplane all the information gathered above.

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MATH 333 – Quiz 4

You may work with other class members on this quiz, but you may not receive assistance from people not in MATH 333 (Section 002). You must show all of your work to receive full credit. Do all your work on other sheets of paper and be sure to staple all the pieces of paper together or YOU WILL GET A ‘ZERO’ ON THE QUIZ. Do not use decimal approximations unless asked to do so. Your work on this quiz must be handed in by Friday, 3 October 2008 at 1040. GOOD LUCK!

  1. Let t > 0. Find the general solution of

dy dt

3 y t

cos 2t t^2

  1. A 100-gallon tank is half full of pristine water. A salt water solution with 5 pounds of salt per gallon enters the tank at 3 gallons per minute and at the same time the well-stirred solution leaves the tank at 4 gallons per minute.

a) Write an initial value problem that models this process. b) Over what time frame does the differential equation model this process? Explain.

c) Solve the initial value problem. d) At what time is the amount of salt in the tank a maximum? Give the exact solution and an appropriate estimate.

  1. Consider the population model

  

 

dR dt

R

R

RF

dF dt

F +

RF

a) Find all equilibrium solutions of (1). b) Describe completely what happens to species F if species R is extinct. c) Describe completely what happens to species R if species F is extinct. d) Depict in the R-F plane all the information gathered above.