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216 Math Study Group 9/18/08 1. Solve the following equations: a. y = 4x3 y; y(1) = -3 b. 3xy + y = 12x 2. The rate of change of the alligator population P(t) is proportional to the square of its size. If there are a dozen gators in 1988 and 2 dozen in 1998, when will there be 4 dozen gators? 3. Suppose the rabbit population satisfies the following equation: dP/dt = aP bP2 where P(0) = 240, B0 = 9, and D0 = 12. At what time does the population reach 105% of the limiting population? ONLY Set up the equation you would use to solve. 4. Find and classify the critical points of the following equations. Then, solve for the general solution. dx/dt = x2 4x *5. Consider the equation dy/dt = (y 5)(y2 1)(y+1). a. Find all equilibrium solutions. Draw b. the phase diagram and classify the equlibria as stable, unstable, or semi- stable. Label your sketch clearly. c. Sketch solution curves (above, below, and in-between the equilibrium solutions) *6. In an investigation, a corpse was discovered with a body temperature of 35 degrees Celsius. Two hours later the body is 30 degrees Celsius. If the temperature of the room is a constant 20 degrees Celsius, how much time elapsed between the murder and the discovery of the body? (Normal body temperature for a living person is 37 degrees C. Newton s law of cooling says that the rate of change of the temperature T(t) of a body is proportional to the difference between T & the temperature of the surrounding medium. It is ok to leave your answer in terms of logs.)
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