Covered today: Basic applications to differential equations. (Sections 2.5 and 2.6)

For next time: Turn in 2.5 - 8,21,27, 2.6 - 3,10,18 (for those without books, these problems are at the bottom of the page.

Transparencies from class:

• Suppose you like to drink your coffee at 95 degrees, so you set a 150 degree cup of coffee outside (where it's 45 degrees). 2 minutes later, the coffee is down to 115. How long until it's at your desired temperature of 95?

• Suppose a 500 gallon tank initially contains 100 gallons of pure water. At time t=0, salt water (consisting of one-half pound of salt per gallon) begins to enter the tank at a rate of 3 gallons per minute. The well-stirred mixture is drained at a rate of 2 gallons per minute. Find an expression for the amount of salt in the tank at any time $t \geq 0$.

2.5 number 8: A young person with no initial capital invests k dollars per year at an annual interest rate r. Assume that investments are made continuously and that interest is compounded continuously.

• Determine the sum S(t) accumulated at any time t.
• If r=7.5%, determine k so that one million dollars will be available for retirement in 40 years.
• If k=2000 dollars per year, determine the interest rate r that must be obtained to have one million dollars available in 40 years.

2.5 number 21: Suppose that a body with temperature 85 degrees is discovered at midnight, and that the ambient temperature is a constant 70 degrees. The body is removed quickly (assume instantly) to the morgue, where the ambient temperature is 40 degrees. After one hour, the body temperature is found to be 60 degrees. Estimate the time of death.

2.5 number 27: Suppose a room containing 1200 cubic ft of air is originally free of carbon monoxide. Beginning at time t=0, cigarette smoke containing 4% carbon monoxide (CO) is introduced into the room at a rate of 0.1 cubic ft per minute, and the well-circulated mixture is allowed to leave the room at the same rate.

• Find an expression for the concentraion x(t) of carbon monoxide in the room at any time t > 0.
• Extended exposure to a CO concentration as low as 0.00012 is harmful. Find the time required until this level is reached.

(You may need to read go back to your calc book to review some definitions for these next two)

2.6 number 3: Consider the DE y'=y(y-1)(y-2). Find all equilibrium solutions, and classify each one as asmptotically stable or unstable.

2.6 number 10: Consider the DE y'=y(1-y²). Find all equilibrium solutions, and classify each one as asmptotically stable, unstable, or semistable.

2.6 number 18: Suppose that the population y of fish in a certain area of the ocean can be modelled by the DE $y' = r(1-\frac{y}{K})y - Ey$, where K and r are positive constants and E is a positive constant (with units 1/time) that measures the effort being made to catch these fish by local fishermen.

• Show that if E < r, there are two equilibrium solutions, y₁ = 0 and y₂ = K(1-E/r) > 0
• Show that y=y₁ is unstable and that y=y₂ is stable.
• A sustainable yield Y of the fishery is a rate at which fish can be caught indefinitely. It is the product of the effort E and the stable population y₂. Find Y as a function of the effort E. This graph is known as the yield-effort curve.
• Determine E so as to maximize Y, thereby finding the maximum sustainable yield Y_m.