I Need help with answering these questions, could you please provide full working out solutions clearly hand written so that i can use it to help me with my next assignment.

. 1. (2+7+3+2+2=16 marks) Consider the initial value problem # + =0, x(0) = 1, r'(0) = 2. (1) (a) Rewrite the second order differential equation as a system of two first order equations. Give the answer in the matrix form. (b) Give the general solution. (c) Determine the specific solution that satisfies the initial conditions. Give the expressions for r(t) and y(t). (d) The initial value problem Eq. (1) is to be solved using the forward Euler method with step h = 2. Argue why this choice of h is or is not plausible. (e) The initial value problem Eq. (1) is to be solved using the backward Euler method. Explain which precautions must be taken so that the numerical solution remains finite as f + 00. . 2. (5+4=0 marks) Consider the initial value problem "! = f(r.v) = zy', (0) =1. (a) Use the modified Euler method with step h = 0.2 to determine the approximate value of the solution at r = 0.2 Give the answer correct to the third decimal place. (b) Use the fourth order Runge-Kutta method with step h = 0.2 to determine the approximate value of the solution at r = 0.2 Give the answer correct to the third decimal place. . 3. (7+2=0 marks) The value of the function f(x) is known at three points f(1) = 1, f(2) = 2, f(4) = 1. (a) Use the idea outlined in Q2 of tutorial 5 to determine an approximate values of the first f'(2) and the second derivatives f"(2) at point s = 2. (b) Which additional information about the function should be given so that the third derivative f"(2) can be estimated? . 4. (8 marks) Consider the boundary value problem day _ dy + ry = 0, '(0) =1. '(2) = =(2). Use central differences with step h = 1 to set up a system of linear equations for the computation of the values for y(0). y(1) and y(2). Give the answer in the form y(0) A y(1) 1(2 ) = B where A is a 3 x 3 matrix and B is a 3 x 1 column vector. DO NOT attempt to solve the system