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1. Find the equilibrium solution(s) to (%) and identify whether they are stable or unstable. Explain why the minus sign in front of the rate constant in (x) is essential in order that the ODE model return physically reasonable results. (Hint: In reality, a hot object is cooling down AND a cool object is heating up.) . The initial temperature of the object is given by u(0) = up and assume T, and k are given constants. Solve the ODE(%) to find an expression for the temperature u(t) at any time in terms of the parameters Ty, ug and k. . Suppose that a person wants to travel with a specific medication that needs to be kept between 2C and 8C. The medication is put in a thermal insulating box. At the start of the trip, the temperature of the medication inside the box is ug = 2C (it has just been taken out of the fridge) and that the temperature of the surrounding (outside the box) is 25C. Assuming that the rate constant is k = 0.052~!, how long can the trip be (in hours) before the medication starts to expire due to heating? . Making the rate constant smaller in question 3 corresponds to making the box a better insulator (and hence more expensive to make). If the trip takes 25 hours, what is the largest rate constant k so that at the end of the trip, the medication is still in the safe temperature interval? . Draw a direction field plot for the ODE (%) corresponding to the parameters T, = 25C and k = 2h~! in the time interval [0,10h] (you can use Matlab, GeoGebra, ...). Include the following solution trajectories in your plot: o the equilibrium solution(s), e the trajectory corresponding to ug = 2C