Provide answers to the questions below

Example 13.2.1. Let P(t) denote the population of a certain country at time t (t 2 0). P = P(t) is known to satisfy the equation: dP = p . P- kp3 dt (t 2 0) where the present population, P(0), is known and p, k are positive constants. (a) By substituting u = P-2, find a first-order linear differential equation for u = u(t). (b) Solve that differential equation by (for example) the integrating factor method. If this method is unfamiliar, please see Appendix A.] (c) Hence solve the original equation for P(t). (d) What is the limit of P(t) as t - co?Fbcalnple 13.3.1. The estimated female population of a large seaside town at ages 78 to 81 last birthday on 1st January 1989 was as follows: Age Estimated last female birthday population 78 2 1050 79 2 19'29 EU 2 1921 81 2 19'19 It is estimated that1 in each of the years 1989 and 1999., there will be a net inward migration of 179 women at each of the ages 73 and 79 last birthday at the date of moving to the seaside town1 and of 299 women at each of the ages 8'0 and 81 last birthday at the date of moving to the seaside town. The mortality rates of women in this town at ages 78 to 81 are as follows: Insurance claims (in f) arriving at an office over the last month have been analysed. The results are as follows: Claim size, c OS c