HI could you help me answer this please?

Section D A body of mass m is suspended by a xed rod of length L. Let k 2 g/L. where g is the acceleration due to gravity. Let g3 be a positive number representing the strength of the air resistance. Ignoring the mass of the rod. the equation governing the angular displacement of the system from the vertical is: (12) dcf) + 53 dtg +ksin3= 0, Answer the following three questions. Each part carries 10 marks. 1. Write this second order equation as a rst order system. and identify the critical points of the system. Show by an appropriate translation of one of the variables that only the two critical points (0, O) and (1r, 0) need be considered. 2. Characterise the behaviour of the system around the critical point (0, 0). 3. Characterise the behaviour of the system around the critical point (a, 0). Draw a diagram of the phase space curves of the actual system. using your knowledge of the behaviour of the linearised systems to demonstrate how points in phase space evolve. Section D The Lotka-Volterra system of equations for variables 33(t) and y(t) is given here, with all constants positive: (1 d (1: = @133 blxy, (1: = a2y + ngy. Answer the following three questions. Each question carries 10 marks. 1. Find the critical points of this system and write down the linearised system at each such point. 2. Find the eigenvalues of the linearised system for each critical point and therefore characterise the behaviour of the system at that point. 3. Show that the trajectories in phase space of the linearised system are closed curves, for the non-zero critical point