Suppose that the population P(t) of a country satisfies

Problem 2P Separate variables and use partial fractions to solve the initial value problems in Problems. Use either the exact solution or a computer-generated slope field to sketch the graphs of several solutions of the given differential equation, and highlight the indicated particular solution.

Problem 1P Separate variables and use partial fractions to solve the initial value problems in Problems. Use either the exact solution or a computer-generated slope field to sketch the graphs of several solutions of the given differential equation, and highlight the indicated particular solution.

Problem 3P Separate variables and use partial fractions to solve the initial value problems in Problems. Use either the exact solution or a computer-generated slope field to sketch the graphs of several solutions of the given differential equation, and highlight the indicated particular solution.

Problem 4P Separate variables and use partial fractions to solve the initial value problems in Problems. Use either the exact solution or a computer-generated slope field to sketch the graphs of several solutions of the given differential equation, and highlight the indicated particular solution.

Problem 5P Separate variables and use partial fractions to solve the initial value problems in Problems. Use either the exact solution or a computer-generated slope field to sketch the graphs of several solutions of the given differential equation, and highlight the indicated particular solution.

Problem 39P Birth and death rales of animal populations typically are not constant; instead, they vary periodically with the passage of seasons. Find P(t) if the population P satisfies the differential equation where t is in years and k and b are positive constants. Thus the growth-rate function r(t) = k + bcos 2?t varies periodically about its mean value k. Construct a graph that contrasts the growth of this population with one that has the same initial value P0 but satisfies the natural growth equation P? = kP (same constant k). How would the two populations compare after the passage of many years?

Problem 6P Separate variables and use partial fractions to solve the initial value problems in Problems. Use either the exact solution or a computer-generated slope field to sketch the graphs of several solutions of the given differential equation, and highlight the indicated particular solution.

Problem 7P Separate variables and use partial fractions to solve the initial value problems in Problems. Use either the exact solution or a computer-generated slope field to sketch the graphs of several solutions of the given differential equation, and highlight the indicated particular solution.

Problem 8P Separate variables and use partial fractions to solve the initial value problems in Problems. Use either the exact solution or a computer-generated slope field to sketch the graphs of several solutions of the given differential equation, and highlight the indicated particular solution.

Problem 10P Suppose that the fish population P(t) in a lake is attacked by a disease at time t = 0, with the result that the fish cease to reproduce (so that the birth rate is ? = 0) and the death rate (deaths per week per fish) is thereafter proportional to 1/ . If there were initially 900 fish in the lake and 441 were left after 6 weeks, how long did it take all the fish in the lake to die?

Problem 9P The time rate of change of a rabbit population P is proportional to the square root of P. At time t = 0 (months) the population numbers 100 rabbits and is increasing at the rate of 20 rabbits per month. How many rabbits will there be one year later?

Problem 11P Suppose that when a certain lake is stocked with fish, the birth and death rates ? and ? are both inversely proportional to ?p. (a) Show that where k is a constant. ________________ (b) If Pq = 100 and after 6 months there are 169 fish in the lake, how many will there be after 1 year?

Problem 12P The time rate of change of an alligator population P in a swamp is proportional to the square of P.The swamp contained a dozen alligators in 1988, two dozen in 1998. When will there be lour dozen alligators in the swamp? What happens thereafter?

Problem 13P Consider a prolific breed of rabbits whose birth and death rates, ? and ?, are each proportional to the rabbit population P = P(t), with ? > ?. (a) Show that k constant Note that P(t) ? + ? as t ? l/(kP0). This is doomsday. ________________ (b) Suppose that P0 = 6 and that there are nine rabbits after ten months. When does doomsday occur?

Problem 14P Repeat part (a) of the Problem below in the case ? < ?. What now happens to the rabbit population in the long run? REFERENCE PROBLEM: Consider a prolific breed of rabbits whose birth and death rates, and , are each proportional to the rabbit population , with . (a) Show that , k constant Note that as . This is doomsday. (b) Suppose that P0 = 6 and that there are nine rabbits after ten months. When does doomsday occur?

Problem 15P Consider a population P(t) satisfying the logistic equation dP/dt = aP — bP2, where B = aP is the time rate at which births occur and D = bP2 is the rate at which deaths occur. If the initial population is P(0) = P0, and Bo births per month and Dq deaths per month are occurring at time t = 0, show that the limiting population is M = B0P0/D0.

Problem 16P Consider a rabbit population P(1) satisfying the logistic equation as in Problem. If the initial population is 120 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?

Problem 18P Consider a population P(t) satisfying the extinction-explosion equation dP/dt = aP2 — bP, where B = aP2 is the time rate at which births occur and D = bP is the rate at which deaths occur. If the initial population is P(0) = Pq and B0 births per month and D0 deaths per month are occurring at time t = 0, show that the threshold population is M = D0P0/Bo.

Problem 19P Consider an alligator population P(t) satisfying the extinction/explosion equation as in Problem. If the initial population is 100 alligators and there are 10 births per month and 9 deaths per months occurring at time t = 0, how many months does it take for P(t) to reach 10 times the threshold population M?

Problem 17P Consider a rabbit population P(t) satisfying the logistic equation as in Problem. If the initial population is 240 rabbits and there are 9 births per month and 12 deaths per month occurring at time t = 0, how many months does it take for P(t) to reach 105% of the limiting population M?

Problem 20P Consider an alligator population P(t) satisfying the extinction/explosion equation as in Problem 18. If the initial population is 110 alligators and there are 11 births per month and 12 deaths per month occurring at lime t = 0, how many months does it take for P(t) to reach 10% of the threshold population M?

Problem 21P Suppose that the population P(t) of a country satisfies the differential equation dP/dt = kP(200 ? P) with k constant. Its population in 1940 was 100 million and was then growing at the rate of 1 million per year. Predict this country’s population for the year 2000.

Problem 23P As the salt KNO3 dissolves in methanol, the number x(t) of grams of the salt in a solution after t seconds satisfies the differential equation dx/dt = 0.8x ? 0.004x2. (a) What is the maximum amount of the salt that will cver dissolve in the methanol? ________________ (b) If x = 50 when t = 0, how long will it take for an additional 50 g of salt to dissolve?

Problem 24P Suppose that a community contains 15,000 people who are susceptible to Michaud’s syndrome, a contagious disease. At time t = 0 the number N(t) of people who have developed Michaud’s syndrome is 5000 and is increasing at the rale 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?

Problem 26P A population P(t) of small rodents has birth rate ? = (0.001) P (births per month per rodent) and constant death rate ?. If P(0) = 100 and P?(0) = 8, how long (in months) will it take this population to double to 200 rodents? (Suggestion: First find the value of ?.)

Problem 27P Consider an animal population P(t) with constant death rate ? = 0.01 (deaths per animal per month) and with birth rate ? proportional to P. Suppose that P(0) = 200 and P'(0) = 2 (a) When is P = 1000? (b) When does doomsday occur?

Problem 25P The data in the table in Fig. are given for a certain population P(t) that satisfies the logistic equation. (a) What is the limiting population M? (Suggestion: Use the approximation with h = 1 to estimate the values of P?(t) when P = 25.00 and when P = 47.54. Then substitute these values in the logistic equation and solve for k and M.) (b) Use the values of k and M found in part (a) to determine when P = 75. (Suggestion: Take t = 0 to correspond to the year 1925.)

Problem 22P Suppose that at time t = 0, half of a “logistic” population of 100,000 persons have heard a certain rumor, and that the number of those who have heard it is then increasing at the rate of 1000 persons per day. How long will it take for this rumor to spread to 80% of the population? (Suggestion: Find the value of k by substituting P(0) and P?(0) in the logistic equation, Eq.)

Problem 28P Suppose that the number x(t) (with t in months) of alligators in a swamp satisfies the differential equation dx/dt = 0.0001x2 ? 0.01x. (a) If initially there are 25 alligators in the swamp, solve tills differential equation to determine what happens to the alligator population in the long run. ________________ (b) Repeat part (a), except with 150 alligators initially.

Problem 29P During the period from 1790 to 1930, the U.S. population P(t) (t in years) grew from 3.9 million to 123.2 million. Throughout this period, P(t) remained close to the solution of the initial value problem (a) What 1930 population does this logistic equation predict? ________________ (b) What limiting population does it predict? ________________ (c) Has this logistic equation continued since 1930 to accurately model the U.S. population? [This problem is based on a computation by Verhulst, who in 1845 used the 1790-1840 U.S. population data to predict accurately the U.S. population through the year 1930 (long after his own death, of course).]

Problem 30P A tumor may be regarded as a population of multiplying cells. It is found empirically that the “birth rate” of the cells in a tumor decreases exponentially with time, so that ?(t) = ?0e??t (where ? and ?0 are positive constants), and hence Solve this initial value problem for Observe that P(t) approaches the finite limiting population P0 exp (?0/?) as t? + ?

Problem 31P For the tumor of Problem 30, suppose that at time t = 0 there are Po = 106 cells and that P(t) is then increasing at the rate of 3 x 105 cells per month. After 6 months the tumor has doubled (in size and in number of cells). Solve numerically for ?, and then find the limiting population of the tumor.

Problem 32P Derive the solution of the logistic initial value problem P? = kP(M ? P), P(0) = p0). Make it clear how your derivation depends on whether 0 < P0 < M or P0 > M.

Problem 34P If P(t) satisfies the logistic equation, use the chain rule to show that Conclude that Pn > 0 if 0 < P < 1/2M;Pn = 0 P = 1/2M; Pn < 0 if 1/2M < P < M; and Pn > 0 if P > M. In particular, it follows that any solution curve that crosses the line P = 1/2M has an inflection point where it crosses that line, and therefore resembles one of the lower S-shaped curves in Fig.

Problem 33P (a) Derive the solution of the extinction-explosion initial value problem P? = kP(P ? M), P(0) = P0. ________________ (b) How does the behavior of P(t) as t increases depend on whether 0 < P0 < M or P0 > M?

Problem 35P Consider two population functions P1(t) and P2(t), both of which satisfy the logistic equation with the same limiting population M but with different values k1 and k2 of the constant k in Eq. Assume that k1 < k2. Which population approaches M the most rapidly? You can reason geometrically by examining slope fields (especially if appropriate software is available), symbolically by analyzing the solution given in Eq., or numerically by substituting successive values of t.

Problem 36P To solve the two equations in for the values of k and M, begin by solving the first equation for the quantity a = e?50km and the second equation for x2 = e?100km. Upon equating the two resulting expressions for x2 in terms of M, you get an equation that is readily solved for M. With M now known, either of the original equations is readily solved for k. This technique can be used to “fit” the logistic equation to any three population values P0, P1, and P2 corresponding to equally spaced times t0 = 0, t1, and t2 = 2t1.

Problem 37P Use the method of Problem to fit the logistic equation to the actual U.S. population data for the years 1850, 1900, and 1950. Solve the resulting logistic equation and compare the predicted and actual populations for the years 1990 and 2000.

Problem 38P Fit the logistic equation to the actual U.S. population data for the years 1900, 1930, and 1960. Solve the resulting logistic equation, then compare the predicted and actual populations for the years 1980, 1990, and 2000.