MATH 3328 Differential Equations
Problem Set 2
Due February 10, 2017
- Professor Rapatski and her husband saw a financial planner to determine how to save for their (first) son’s college. He suggested they cash in some stocks worth $20,000 to start the college fund and then deposit an additional $200 a month ($2400 a year). They will put the money in a NJ Best 529 College Savings plan gaining 8% interest a year. Assume interest and deposits are continuous.
a) Write a differential equation to represent the amount in the college fund at time t with an initial condition. Solve and determine how much money there will be in the savings account when the child is ready to start college (in 18 years assuming they started saving from child’s birth).
b) Instead of opening the account with $20,000 Professor Rapatski and her husband opened the account with $6500 and used the rest of the money to put in an in-ground pool (true story). They decided that they will contribute an extra $200 a month until the $13,500 is paid back. So for 5.625 years they will contribute $400 each month. After which they will return to depositing just $200 a month. How much money will the college savings account have after 18 years?
c) Why are the answers not the same? By getting the pool, how much less is the college fund?
2. Prove the initial value problem, has one and only one solution. (Note: You do not need to actually compute the solution). Moreover, prove that this solution satisfiesfor all t for which it is defined. Conclude that the domain of is the whole line. Include accurate pictures illustrating your reasoning.
a) Find the bifurcation value(s).b) Sketch the bifurcation diagram.
- At Herculane (a spa in southwestern Romania, named after the legendary Hercules) a pool for rheumatic treatment, when half filled, contains 50,000 gallons of natural hot spring-water that has 5000 pounds of Herculane salt. Fresh water is poured into the pool at the rate of 2000 gallons per hour. Assuming that the fresh water mixes instantly with the hot spring water and that the solution leaves the tank at a rate of 1000 gallons per hour, what is the salt concentration when the tank is completely filled?
- Let . The graph of is given below:
a) Sketch the phase line when.
b) Describe the bifurcations that occur in the one parameter family and sketch the bifurcation diagram.
- Find the general solution and the solution that satisfies the initial condition.
- Solve the initial value problem