A dose of morphine is absorbed from the bloodstream of a patient at a rate proportional to the amount in the bloodstream. (a) Write a dierential equation for M????t, the amount of morphine in the patients bloodstream, using k as the constant proportionality. (b) Assuming that the initial dose of morphine is M0, solve the initial value problem to nd M????t. Use the fact that the half-life for the absorption of morphine is two hours to nd the constant k. (c) Suppose that a patient is given morphine intraveneously at the rate of 3 milligrams per hour. Write a dierential equation that combines the intraveneous administration of morphine with the bodys natural absorption. (d) Find any equilibrium solutions and determine their stability. (e) Assuming that there is initially no morphine in the patients bloodstream, solve the initial value problem to determine M????t. What happens to M????t after a very long time? (f) To what rate should a doctor reduce the intraveneous rate so that there is eventually 7 milligrams of morphine in the patients bloodstream?