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**Extra resources for Introduction to Ordinary Differential Equations. Second Enlarged Edition with Applications**

**Sample text**

3. An object whose temperature is 220°F is placed in a room where the temperature is 60°F. After 10 min the temperature of the object is 200°. At this point refrigeration equipment, which lowers the temperature of the room at the rate of l°F/min, is turned on. What is the temperature of the object t min after the equipment is turned on ? 4. An object with a temperature of 10°F is placed in a room where the temperature is 80°F. After 10 min the temperature of the object is 30°. What will be the temperature of the object after it has been in the room for 30 min?

C) If F is as in part (b), find a formula for the quantity k in part (a). 9 COOLING: THE RATE OF A CHEMICAL REACTION The first problem that we consider in this section has to do with the change in temperature in a cooling body. If a body cools in a surrounding medium (such as air or water) it might be expected that the rate of change of the temperature of the body would depend on the difference between the temperature of the body and that of the surrounding medium. Newton's law of cooling asserts that the rate of change is directly proportional to the difference of the temperatures.

2. Using the data of Exercise 1, find the population as a function of time, assuming that the model dNjdt = kN1/2 applies. 3. (a) Find a formula for N(t) if JV(0) = N0 and dN 1 -*=kN> where a is a positive constant, α Φ 1. (b) If 0 < a < 1 in part (a), what happens to N(t) as t becomes infinite? (c) If a > 1 in part (a), show that N(t) becomes infinite after a finite time. 4. 39) of Eq. 38). 5. Initially the population of a species is 5000. After 10 days it is 8000. After a very long time the population stabilizes at 15000.