Here is a somewhat realistic example you should use the

Question

Here is a somewhat realistic example. You should use the phase plane plotter to look at some solutions graphically before you start solving this problem and to compare with your analytic answers to help you find errors. You will probably be surprised to find how long it takes to get all of the details of solution of a realistic problem right, even when you know how to do each of the steps.

There are 2000 dollars in the bank account at the beginning of January 1990, and money is added and withdrawn from the account at a rate which follows a sinusoidal pattern, peaking in January and in July with money being added at a rate corresponding to 2100 dollars per year, while maximum withdrawals take place at the rate of 360 dollars per year in April and October.

The interest rate remains constant at the rate of 5 percent per year, compounded continuously.

Let y(t) represents the amount of money at time t (t is in years).

y(0)=_____________________________(dollars)

Write a formula for the rate of deposits and withdrawals (using the functions sin(), cos() and constants):

g(t) = ________________________________

Hint: Write it in the form: A+Bcos(ωt+?).

The interest rate remains constant at 5 percent per year over this period of time.

With y representing the amount of money in dollars at time t (in years) write a differential equation which models this situation.

y′=f(t,y)= ____________________________.

Note: Use y rather than y(t) since the latter confuses the computer. Don't enter units for this equation.

Find an equation for the amount of money in the account at time t where t is the number of years since January 1990.

y(t)= _________________________________

Find the amount of money in the bank at the beginning of January 2000 (10 years later): _____________________

Find a solution to the equation which does not become infinite (either positive or negative) over time:

y(t)= _____________________________________

During which months of the year does this non-growing solution have the largest magnitude?   January and July    February and August    March and September    April and October    May and November    June and December