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To solve the separable differential equation dy + ycos(x) = 9 cos(x), we must find two separate integrals: dy = and Solving for y we get that y = (you must use k as your constant) and find the particular solution satisfying the initial condition y(0) = -8. y(I) =A culture of yeast grows at a rate proportional to its size. If the initial population is 1000 cells and it doubles after 2 hours, answer the following questions. 1. Write an expression for the number of yeast cells after t hours. Answer. P(t) = 2. Find the number of yeast cells after 9 hours. Answer. 3. Find the rate at which the population of yeast cells is increasing at 9 hours. Answer (in cells per hour):In many population growth problems, there is an upper limit beyond which the population cannot grow. Man},r scientists agree that the earth will not support a population ot more than 16 billion. There were 2 billion people on earth at the start ot1925 and 4 billion at the beginning of 1975. if y is the population. measured in billions. t years after 1925, an appropriate model is the differential equation do =k 15 . a\" at a} Note that the growth rate approaches zero as the population approaches its maximum size. When the population is zero then we have the ordinary exponential growth described by y' = 16kg. As the population grows it transits from exponential growth to stability. {at Solve this dierential equation. y = I I (bi The population in 2015 will be 3,: =[ lbillion. {cl The population will be 9 billion some time in the year| ] Note that the data in this problem are out of date, so the numerical answers you'll obtain will not be consistent with current population figures. Hint: (at Separate variables and use the given information to solve for y. (b) Evaluate y. (c) Solve forthe appropriate time. Find the solution to the initial value problem (1 + x)y' + 91By = 2x12 subject to the condition y (0) = 2.A tank initially contains 200 gallons of brine, with 50 pounds of salt in solution. Brine containing 2 pounds of salt per gallon is entering the tank at the rate of 4 gallons per minute and is is flowing out at the same rate. If the mixture in the tank is kept uniform by constant stirring, find the amount of salt in the tank at the end of 40 minutes. The amount of salt in the tank at the end of 40 minutes is pounds.For this problem A is the amount of salt in the tank. If a tank contains 250 liters of liquid with 11 grams of salt. A mixture containing 10 grams per liter is pumped into the tank at a rate of 6 liters/minute. The mixture is well-mixed, and pumped out at a rate of 9 liters/minute. The amount of salt in the tank satisfies the differential equation = 0. Rewriting this as a linear differential equation we get 60 The integrating factor is and the solution is