Recall that Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference in temperature between the object and the surrounding medium. This leads dT to the differential equation shown below. = HT Tm), where T is the temperature of the alt object after time t, TM is the temperature of the surrounding medium, and k is a constant. a) Is the equation linear? b) Find the solution to the linear equation using the method for solving a linear first-order differential equation. of First, rewrite the equation in the linear form dy + P(m]y = Q[m). :1: GET a + '\"T = l l Next, find P(t] and Q(t). P(t) = 1 ext) =1: 1 Find the integration factor. fit) = li Now multiply each term of the equation by the integrating factor. The LHS is given. Find the RHS. (stag + (Jaw = 1 1 Replace the sum of the term on the left with DI[I[m)y], or Dt[I[t)T], treating t as 9:: and T as y. Dill. l ] = i'hkzkif Set up the integration. Then integrate on both sides of the equation. [Dr [Tekt] dt = [kaektdt Integrate the left side. i l = [kaektdt Next integrate the right side of the equation. T ekt = 11 Solve the equation for T. T=l fl