Answer the following question. Follow the format given and write legibly to avoid confusions of answer. THANK YOU VERY MUCH!!

1.

Find the general solution of the given differential equation, and use it to determine how solutions behave as t - co. + = 3 cos(4t), t> 0 t. NOTE: Use c for the constant of integration. y Solutions converge to the function y =Solve the initial value problem. 2 ,y(0)=v Let be the value of a for which the transition from one type of the 3y'2y=e long-run behavior to another occurs. Find the critical value (1.3 exactly. Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 160 F when freshly poured, and 1 min later has cooled to 150 F in a room at 70 F, determine when the coffee reaches a temperature of 120 F. Note: Enter the exact symbolic expression for the asked time. The coffee reaches the temperature 120 in minutes.Suppose that a certain population has a growth rate that varies with time and that this population satisfies the differential equation dy dt = (0.4 + sint) NOTE: Round your answers to two decimal places. a) If y(0) = 1, find (or estimate) the time 7 at which the population has doubled. Choose other initial conditions and determine whether the doubling time ~ depends on the initial population. The doubling time is T = The doubling time Choose one depend on the initial population.b) Suppose that the growth rate is replaced by its average value 10 Determine the doubling time 7 in this case. In this case, T = c) Suppose that the term sin t in the differential equation is replaced by sin (27t); that is, the variation in the growth rate has a substantially higher frequency. What effect does this have on the doubling time ? In this case, T =