Math 3328, Spring 2016

You must show necessary work to get full credit! No notes, but calculators allowed!

Name (2pts):______

  1. (13 pts) Suppose your town has a small lake that is stocked with fish. Assume the fish population can be modeled by a logistic model with a carrying capacity of 50 and a growth rate of 2.

a)  Write the equation of the model.

b)  Suppose it is decided that fishing will be allowed. Every person interested in fishing must get a fishing license and the average catch per licensed fisherman is 3 fish per year. Write the new model including the fishing.

c)  What is the largest number of licenses that your town can issue so the fish have a chance to survive in the lake? Justify completely.

Find equilibrium solutions:

If L>25/3, no equilibrium solutions (solutions would always be decreasing)

If L=25/3, one equilibrium solution, P=25. On the phase line solutions will decrease above P=25 and below P=25.

If L<25/3, two equilibrium solutions. On the phase line solutions decrease above the larger equilibrium, increase in between and decrease below the lower equilibrium.

The only chance for the population to die out is L<25/3, so L=8 is the max. As long as the population starts out large enough, the population will not go extinct.

  1. (12 pts each) Solve the initial value problems.

a) 

b) 

Hom:

Particular:

  1. Assume . The following is the graph of, and has three roots.

a)  (4pts) Sketch the phase line and classify the equilibrium points.

1 = sink, 0 = node, -1 = source

b)  (2pts) Sketch the graph of the solution satisfying the initial conditions

Curve goes to 1 as and goes to 0 as

c)  (8pts) Describe the bifurcation(s) that occur in the one parameter family. Sketch the bifurcation diagram.

The number of equilibrium solutions changes based on alpha.

Sketch the phase line for each case. Bifurcation values are 0 and -2.5. When you connect, it looks like a sideways W.

  1. (12 pts) At a wine cooler plant there is a large tank initially holding 100 gallons of a juice mixture (no alcohol). Pure alcohol enters at a rate of 4 Gal/min and is well stirred. The mixture leaves the tank at a rate of 2gal/min.

a)  Write a differential equation to represent the amount of alcohol in the tank at time t.

b)  Wine coolers are supposed to be no more than 7% alcohol. The plant manager lets alcohol pour in for 5 minutes and then stops and begins to bottle the mixture. What is the percent alcohol after 5 minutes? Can the wine coolers be sold?

Integrating factor

Can’t be sold

  1. (5 pts) . Prove the solution to the initial value problem is unique and satisfies.

are both continuous. By the Existence/Uniqueness Theorem, there exist a unique solution. The only equilibrium solution is y=0. By uniqueness, solutions cannot cross equilibrium solutions. Thus since y(0)=2>0 then y(t)>0 for all time.

  1. (8 pts) One of the houses on the TV show Hoarders has a large mouse infestation. The mouse population has a growth rate of 5 mice per week. Mice can be modeled by the unlimited growth population model. The clean-up crew is on a mission and sets up humane traps everywhere capturing 300 mice a day (which get sent to a pet store far-far away). The population of mice can be modeled by:

After just four days the clean-up crew claim they have captured all the mice. How many were there initially?

  1. (8 pts) . Use Euler’s method with step size to approximate
  1. (14 pts) As you already know Professor Rapatski and her husband would have saved $159,931.31 for their son Joey’s college if they followed the financial planner’s advice. What they really did was open up a NJ Best 529 College Savings plan with an initial deposit of $6500 when Joey was one-year old. The savings plan gains 8% interest a year. They plan to contribute M dollars each month (12M a year). Assume interest and deposits are continuous.
  2. Write a differential equation to represent the amount in Joey’s college fund at time t with an initial condition.
  1. Solve the differential equation. (Note: your solution will have M in it).

  1. How much money each month should Professor Rapatski and her husband contribute to reach $159,931.31 for Joey’s college? Note: Joey will start college when he’s 18 years old.