15) The population P of some rare rabbits in a small island is given by: dP where M is the carrying capacity. dt = kP(M-P) M represents the maximum sustainable number of rabbits in the island, and k is a constant which may be positive or negative, based on whether the number of rabbits grows or decays. (a) The island has an initial population of 200 rabbits, and after one year, the population increases to 400 rabbits. Solve this differential equation in the case in which the island has a carrying capacity of 1500. Give the answer in terms of time / only. (b) What is the rabbit population after 5 years? 16) Consider the initial value problem: dy --2x-y=0 dx with y(0) = 1 a) Solve this IVP exactly using any method of your choice. b) Calculate y(1) to four decimal places. Then determine y(1) to four decimal places numerically: c) using the Euler method with h= 0.1 d) using the fourth order Runge -Kutta method with h= 0.1 using the second order Runge -Kutta method with h= 0.1 Tabulate your results, and show at least one sample set of calculations for each method.