(I know it's long but would be very much appreciated if someone could answer this question)

This question will investigate the solution to a coupled system of differential equations and how to transform it to a system that can be solved by the eigenvector method. It is desired to solve the coupled system of differential equations = x + 2y - 50 2x+y- 40 where x and y represent the population of two types of symbiotic coral andt is time measured in decades. [2 marks) 2a. Find the equilibrium point for this system, 2b. If initially r = 100 and y=50 use Euler's method with an time [3 marks) increment of 0.1 to find an approximation for the values of randy when t = 1. [2 marks] 20. Extend this method to conjecture the limit of the ratio ast+00. [3 marks) 2d. Show how using the substitution X = 2 10, Y = y - 20 transforms I X = X+2Y the system of differential equations into Y = 2X+Y 2e. Solve this system of equations by the eigenvalue method and hence find /9 marks) the general solution for of the original system. - (3) [2 marks) 2f. Find the particular solution to the original system, given the initial conditions of part (b). 29. Hence find the exact values of a hnd y when t = 1, giving the answers to[2 marks] 4 significant figures. 2h. Use part (1) to find limit of the ratio as t+0. [3 marks] 2i. With the initial conditions as given in part (b) state if the equilibrium point [1 mark) is stable or unstable. This question will investigate the solution to a coupled system of differential equations and how to transform it to a system that can be solved by the eigenvector method. It is desired to solve the coupled system of differential equations = x + 2y - 50 2x+y- 40 where x and y represent the population of two types of symbiotic coral andt is time measured in decades. [2 marks) 2a. Find the equilibrium point for this system, 2b. If initially r = 100 and y=50 use Euler's method with an time [3 marks) increment of 0.1 to find an approximation for the values of randy when t = 1. [2 marks] 20. Extend this method to conjecture the limit of the ratio ast+00. [3 marks) 2d. Show how using the substitution X = 2 10, Y = y - 20 transforms I X = X+2Y the system of differential equations into Y = 2X+Y 2e. Solve this system of equations by the eigenvalue method and hence find /9 marks) the general solution for of the original system. - (3) [2 marks) 2f. Find the particular solution to the original system, given the initial conditions of part (b). 29. Hence find the exact values of a hnd y when t = 1, giving the answers to[2 marks] 4 significant figures. 2h. Use part (1) to find limit of the ratio as t+0. [3 marks] 2i. With the initial conditions as given in part (b) state if the equilibrium point [1 mark) is stable or unstable