Application (40 pts): In this problem, we want to predict the population of rabit P(t) over time. We model the evolution of the population using the following initial value problem:

where: r is the growth rate M is the maximum population or carrying capacity P0 is the initial population

For this problem, we consider the following parameters: r = 0.4, M = 1000, P0 = 200.

(a) (20 points) Using each of the methods implemented in the previous question, approximate the solution of the IVP at time t = 6 using 10 timestep

(b) (10 points) This problem can actually be solved analytically and the exact solution is given by:

For each of the methods, what is the absolute error at the final time for the approximations you got in question (a)? Which method is more accurate?

(c) (10 points) Instead of using 10 timesteps, recompute the approximation and the absolute error using 100 steps. Using this value, compute the order of convergence of each of the methods. Do you get the expected order of convergence?