In MATLAB can you write this code

3. Pursuit problem. Suppose a rabbit follows a predefined path r(t) = (1.1 cos(t),sin(t)). A fox, initially at y(0) = (0, 0), chases the rabbit in such a way that at each moment the fox the runs toward the rabbit with a constant speed k. That is, the path of the fox, y(t), is determined by the ODE dy(t) dt = k r(t) ? y(t) kr(t) ? y(t)k , where k k denotes the length (i.e. 2-norm) of a vector. Obviously, this equation is ill-defined if the distance kr(t) ? y(t)k = 0. Thus we consider that the fox captures the rabbit if this distance is very close to 0.

(a) Develop a script (P3.m) and necessary functions to solve this problem. Set the events function properly such that the solver ode45 terminates once the distance kr(t) ? y(t)k reaches 10?5 . To avoid the failure of the ode solver, you also need to set other options (eg. error tolerance and/or step size) properly.

(b) Run your code with k = 1.1 and find the time instant when capture occurs. Generate a graph showing the trajectories of the rabbit and the fox. Add the initial positions and where the rabbit is captured as symbols. Properly annotate your plot.

(c) The fox is not supposed to capture the rabbit if k ? 1. Verify this by running your code with k = 0.9 for t ? 20. Generate a graph showing the trajectories.