Differential Equations Problem

Name __________________________ Differential Equations Problem Set Score ______ Show all work neatly done in pencil, or typed, if possible. 1. Solve the initial value problem. a) kP dt dP = 100 P0 = b) 3 4 1 dx y dy = y(0) =1 c) = 2y − 3 dx dy y(0) = 2 d) x( ) x dt dx = 3 5 − x(0) = 8 2. In a certain culture of bacteria, the number of bacteria increased tenfold in 10 hours. Assuming natural growth, how long did it take for their number to double? 3. The half-life of radioactive cobalt is 5.27 years. Suppose that a nuclear accident has left the level of cobalt radiation in a certain region at 100 times the level acceptable for human habitation. How long will it be before the region is again habitable? 4. In 1998 there were 40 million Internet users in the world and this number was then doubling every 100 days. Assuming that this rate of growth continued, how long would it be until all the world’s 6 billion human beings were using the Internet? 5. The population of a certain prolific breed of rabbits satisfies the initial value problem 2 kP dt dP = . Initially there are only 2 rabbits (a male and a female) and after 3 months there are 4 rabbits. How many rabbits will there be after 6 months? 6. Consider a rabbit population P(t) satisfying the logistic equation 2 aP bP dt dP = − , where B = aP is the time rate at which births occur and 2 D = bP is the rate at which deaths occur. If the initial population is 240 rabbits and there are 8 births per month and 6 deaths per month occurring at time t = 0 , how many months does it take for P(t) to reach 95% of the limiting population M? 7. If a Modern pizza is cooling in a medium with a constant temperature A, then according to Newton’s Law of Cooling the rate of change of temperature T of the pizza is proportional to T – A. We plan to cool the pizza initially at C o 80 by setting it on the front porch, where the temperature is . 0 C o If the temperature of the pizza drops to C o 50 after 10 minutes, when will it be C o 40 ( F) o 104 and ready to eat? 8. Just before midday the body of an apparent homicide victim is found in a room that is kept at a constant temperature of . 70 F o At 12 noon the temperature of the body is F o 80 , and at 1pm it is . 75 F o Assume that the temperature of the body at time of death was F o 98 and that it has cooled in accordance with 6. Newton’s law of cooling. What was the time of death? Whodunnit??? 9. Orange, CT. has a fixed population of 10,000 people. On January 1, 1000 people have the flu. On April 1, 2000 people have the flu. Assume that the rate of increase of the number N(t) who have the flu is proportional I the number who don’t have it. How many will have the disease on October 1? 10. Suppose that the population P(t) (in millions) in Mauritania satisfies the differential equation kP( ) P dt dP = 200 . Its population in 1960 was 100 million and w − as growing at a rate of 1 million per year. Predict the country’s population in 2010. Extra Credit 1. Suppose that a community contains 15,000 people who are susceptible to Michaud’s syndrome, a contagious disease. At time 0 t = the number N(t) of people who have developed Michaud’s syndrome is 5000 and is increasing at the rate of 500 per day. Assume that N′(t) is proportional to the product of the numbers of those who have caught the disease and of those who have not. How long will it take for another 5000 people to develop Michaud’s syndrome?