Consider the differential equationdPdt= kP1 + c, wherek > 0andc ≥ 0.In Section 3.1 we saw that in the casec = 0the linear differential equationdP/dt = kPis a mathematical model of a population P(t) that exhibits unbounded growth over the infinite time interval[0, [infinity]),that is, P(t) → [infinity]ast → [infinity].See Example 1 in that section.(a) Suppose forc = 0.01that the nonlinear differential equationdPdt= kP1.01, k > 0,is a mathematical model for a population of small animals, where time t is measured in months. Solve the differential equation subject to the initial conditionP(0) = 10and the fact that the animal population has doubled in 6 months. (Round the coefficient of t to six decimal places.)P(t) =(b) The differential equation in part (a) is called a doomsday equation because the population P(t) exhibits unbounded growth over a finite time interval(0, T),that is, there is some time T such thatP(t) → [infinity]ast → T −.Fin