16. The graph of f(y) is given below, use it to answer the following. f 2 a b y (a) Sketch the phase line for y' = f(y). Identify each equilibrium as stable, unstable, or semi- stable. (b) For what values of k does the equation y' = f(y) - k have two stable equilibria?
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- An antibiotic is administered intravenouslyinto the bloodstream at a constant rate r. As the drug flowsthrough the patient’s system and acts on the infection that is present,it is removed from the bloodstream at a rate proportional tothe amount in the bloodstream at that time. Since the amount ofblood in the patient is constant, this means that the concentrationy = y(t) of the antibiotic in the bloodstream can be modeled bythe differential equation dy/dt = r - ky, k > 0 and constant. If y(0) = y0, find the concentration y(t) at any time t.FIND THE DERIVATIVES NO.1 AND NO. 2Assume that the dynamics of caffeine absorption are given by Ct+1 = 0.87Ct , where t is time in hours and Ct is the concentration of caffeine. If C0 = 1000, estimate the time needed for 80% of the caffeine to be eliminated from the body (i.e. 20% left). What if C0 = 500?
- 2. Determine the derivatives of the ff. functions and show solutionA spring has a mass of 10 kg, is underdamped with coefficient β = 5 and its frequency is 0.8 Hz. Spring starts 30 cm from equilibrium position and is released with a velocity of −10 cm / s. Set up, in terms of the spring constant k, an equation and conditions for the position of the mass, in meters.Consider a pond that initially contains 10 million gal of fresh water. Water containing an undesirable chemical flows into the pond at the rate of 5 million gal/yr, and the mixture in the pond flows out at the same rate. The concentration y(t) of chemical in the incoming water varies periodically with time according to the expression y(t) = 2 + sin 2t g/gal. Construct a mathematical model of this flow process and determine the amount of chemical in the pond at any time.