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1. Solve the given initial value problem. dy/dt = (t^3 + t)/(y^2); y(0) = 2 ....

1. Solve the given initial value problem. dy/dt = (t^3 + t)/(y^2); y(0) = 2 .

2. We know from Newton’s Law of Cooling that the rate at which a cold soda warms up is proportional to the difference between the ambient temperature of the room and the temperature of the drink. The differential equation corresponding to this situation is given by y' = k(M − y) where k is a positive constant. The solution to this equation is given by y = M + (y0 − M)e^−kt, where y0 is the initial temperature of the soda.

a) Suppose the room is held at a steady temperature of 70 degrees Fahrenheit, a cold soda is initially 45 degrees Fahrenheit, and after 2 hours the soda’s temperature has risen to 60 degrees Fahrenheit. Find the values of k, M, and y0 for this situation.

b) Using your answers from above, when will the soda reach 68 degrees Fahrenheit?

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