 # Differential Equations Questions

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Question 5 (20 marks) Suppose that a drug is administered once every 4 hours, where the drug moves from the gastrointestinal tract into the circulatory system. Denote by c(t) the dosage at time t and by a and b the rates at which the drug is consumed in the gastrointestinal tract (GT) and the circulatory system (CS) respectively. Initially, neither the GT nor the CS contain the drug, so that the situation can be modelled as dx = dt an = =ax-by,y(0)=0 dt Co Ost<1 Suppose that c(t) = 0 1c(++4)=c(t). (a) Plot c(t). (b) Show that over one day (ie 0 t < 24) we have - and 6 = (c) Use the Laplace transform to (d) Use the Laplace transform and to For the following questions assume that co=2. (e) Now suppose that the amount of drug present in the GT can be modelled via an exponential decay function with half life of 2 hours, so a that a= 122 In 2 2 and that the half life in the CS is 4 hours, so that b IN2 Describe what happens to the drug concentration in the GT and in the CS. (f) What happens if the half-life in the GT is 1 hour and the half-life in the CS is 4 hours? Describe how this change affects the concentration in the CS. (g) What happens if the half-life in the GT is 1 hour and the half-life in the CS is 5 hours? Describe how this change affects the concentration in the CS. Hint: In considering (e) to (f) a plot of the solutions will be useful.

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