1. (a) Verify that the function y = x³ 2xy + 2y = 8x³ (b) Find a value for c that satisfies the initial condition y(2) = 6. 2. Use separation of variables to solve the following problems dy =-ty, y(0)=1/√ dt y² + 5 y (a) (b) dy dt -, y(0) = -2 dy (c) = 3y(y - 5), y(0) = 8 dt where 3. Consider the following very simple model of blood cholesterol levels based on the fact that cholesterol is manufactured by the body for use in the construction of cell walls and is absorbed from foods containing cholesterol: Let C(t) be the amount (in milligrams per deciliter) of cholesterol in the blood of a particular person at time t (in days). Then is a solution to the differential equation dC dt = k₁(NC) + k₂E N = person's natural cholesterol level k₁ production parameter E k₂ daily rate at which cholesterol is eaten absorption parameter (a) Suppose N = 200, k₁= 0.1, k₂ = 0.1, E = 400, and C(0) = 150. What will the person's cholesterol level be after 2 days on this diet? (Hint: solve the ODE, then plug in t = 2) (b) With the initial conditions as above, what will the person's cholesterol level be after 5 days on this diet? (c) What will the person's cholesterol level be after a long time on this diet? (d) High levels of cholesterol in the blood are known to be a risk factor for heart disease. Suppose that, after a long time on the high cholesterol diet described above, the person goes on a very low cholesterol diet, so E changes to E = 100. (The initial cholesterol level at the starting time of this diet is the result of part (c).) What will the person's cholesterol level be after 1 day on the new diet, after 5 days on the new diet, and after a very long time on the new diet? 1

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