Math 220

Prof. Mary Beth Hampshire

Submitting Take Home Assignments

General Guidelines:

When submitting problems to be graded, adhere to the following guidelines:

  • Be organized and neat
  • Show all your work!
  • I would prefer that you show your work and answers on the assignment sheet. However, use loose leaf paper (no edges) if needed and staple or paper clip any additional pages to the assignment sheet
  • Indicate the number of the problem beside relevant work with your answer clearly indicated

You may work together on the assignment, but the write-up of the solutions should be your own (not a copy of someone else’s work! This includes web sites, solution guides, Maple, Wolfram Alpha, etc.). Also, as a general rule of thumb, you should not submit something as your work on a take-home assignment unless you are prepared to explain the work and solution to your instructor and/or the class. You may also seek help from your instructor. Because I do allow you to seek help on the take home assignments, I do expect the work to be correct and complete; the assignment will be graded accordingly.

Refer to the syllabus for information regarding late penalties.

Math 220

Take Home Assignment #1

Due: Beginning of class, Thursday September 22

Show ALL your work! Each question is worth 4 points unless otherwise indicated.

  1. (2 points)

a) Show that is a solution to the following

b) Explain why the equation has no real-valued solution.

  1. is a two parameter family of solutions to the second-order DE . Find a solution of the second-order IVP consisting of this DE and the following initial conditions: .

  1. (8 points) Consider the differential equation: for the population p (in thousands) of a certain species at time, t. Note: this type of DE is referred to as an autonomous first order DE (i.e. a DE of the form ).

a)Identify the order, degree and linearity of the DE.

b)What method could be used to solve the DE analytically? Do NOT solve, just state the method for solving including the integration technique required.

c)Produce a sketch of the direction field using Maple and attach the output to this assignment sheet. Use the direction field to answer parts d-g.

d)Find the equilibrium solutions (i.e. constant solutions of form ) analytically and identify these solutions on the direction field.

e)If the initial population is 4000 (i.e. p(0) = 4), what can you say about the limiting population,

f)If the initial population is 1700, what is

g)If the initial population is 800, what is

h)Can a population of 900 ever increase to a value of 1100 (according to the model)? Explain.

For problems 4-7, solve the DE. Unless the explicit solution is specifically requested, you may leave your solution in implicit form.

  1. 5. (explicit form)

6. 7. (explicit form)

For problems 8 & 9, solve the IVP. Give the explicit form for each.

8. 9.

Applications and Modeling:

  1. A cup of hot coffee initially at 95˚C cools to 80˚C in 5 minutes while sitting in a room with a thermostat readingof 21˚C. Use Newton’s law of cooling to determine when the temperature of the coffee will be 50˚C.

  1. The population of a town grows at rate proportional to the population present at time, t. The initial population of 500 increases by 15% in 10 years. What will the population be in 30 years? How fast is the population growing at t = 30?
  1. A 100-volt electromotive force is applied to an RC series circuit in which the resistance is 200 ohms and the capacitance is farad. Find the charge on the capacitor if q(0) = 0. Also, find i(t).