2. a. The normal body temperature of a cat is 40°C. A cat is hit by a car at some time during the night. When it is discovered at 6 AM, the young scientist discovers that the body temperature of the cat is 31°C (T(0) = 31). For this problem, t = 0 corresponds to 6 AM. The early morning temperature is estimated to be 15°C. Newton's Law of Cooling with a constant environmental temperature (T_e = 15) gives the differential equation:

where T(t) is the body temperature of the cat. Find the solution to this differential equation.

b. Two hours later the temperature of the body is found to be 29°C. Find the constant of cooling, k₁, in the differential equation and determine when the death occurred.

c. Since it is early in the morning, the temperature has been decreasing for some length of time rather than remaining constant. Suppose that a more accurate cooling law is given by the differential equation

where T_e is the same as in Part a., but k₂ is a new constant to be determined. Use Maple's dsolve to find the solution of this differential equation. Find k₂ using either Newton's method (Excel sheet) or Maple's fsolve routine. Determine the approximate the time of death using this differential equation and compare to the previous solution. Graph the solutions to both of these differential equations for -6 < t < 2 (from midnight to 8 AM).