In other words, what kind of population do you think it describes? [Actually solving the equation is not helpful. ] 5. Write down a di? erential equation for the velocity v(t) of a falling body of mass m if air resistance is proportional to the square of the instantaneous velocity. (Remarks: Consider the forces acting on a falling object, and what they must add up to by Newton’s Second Law. You do not need to solve the di? erential equation. ) 2 2 6. The ? u makes its ? rst appearance in Pleasant Island (population 999) when Mrs.
Smith’s niece arrives for a visit not feeling well. Let y(t) be the number of infected people. Suppose that apart from the initial visitor, no one enters or leaves the island for an inde? nite period of time. Suppose also that the number of islanders infected grows at a rate proportionate both to the number of people infected and the number not infected (i. e. the number of infections is proportionate to the product). Further suppose that the number of infections triples daily. (a) Write an initial value problem odeling the situation for y(t) the number of people infected at time t. (b) Solve the IVP. 7. A large tank is partially ? lled with 100 gallons of ? uid in which 10 pounds of salt is dissolved.
Brine containing 1 2 pound of salt per gallon is pumped into the tank at a rate of 6 gal/min. The well-mixed solution is then pumped out at a slower rate of 4 gal/min. Find the concentration of the solution in the tank after 30 minutes. 8. (a) What property of the di? erential equation v 1 ? t y + y = tan t guarantees that a family of solutions exists? Answer this in one word. ) v (b) Does a unique solution to the IVP 1 ? t y + y = tan t, y(0) = 4 exist? (c) If a unique solution to the IVP does exist, what is the largest interval over which it exists?