3'. [3 marks] Find an equation of a curve that passes through the point [1, 2) with the propertyr that the slope of the curve at any point (I. y} is equal to 13!]; Give your answer in the form 3; = flirt}. E. [3 marks] Find the solution to the differential equation x2y + dyfdx = yfx for which y) = Re. Give your answer in the form 3: = flix]. 9. [12 marks] A tank contains lfl litres of pure water. Brine that contains UL] kg of salt per litre enters the tank at a rate of ll] LImin. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 6 minutes? ll]. [12 marks] a. {E marks] Show that, if P is a twice differentiable function of t and satises the logistic differential equation at? P he? =ke[1}=ke , dt M M then d2? _ kzP 1 2P 1 P F ( _F)l _H)' l1 {'5 marks] Suppose a population grows according to the logistic growth model and that the initial population size is positive but less than half the carrying capacityr of the environment. Use the result from part a. of this question to show the growth rate of the population is maximal when the population size is exactly half the carrying capacity