Consider the following initial value problem for the function x(t): "(t) + x(t) = 0, x(0) =1, I'(0) = 0. (a) Introduce an auxiliary variable y = ' and recast this ODE into a first-order system of ODEs. (b) Using trigonometric identities, show that the true solution satisfies ( y(+ + At ) ) = (D) (CB ). A(At ) : = os(At) sin(At) - sin(At) cos(At) (c) Now fix At E (0, 1) and let us approximate the true solution with two sequences un, Un such that (un, Un) is an approximation of (x(nAt), y(not)). Write down the iteration schemes for (i) Forward Euler, (ii) Backward Euler, (iii) Crank-Nicolson and (iv) Heun associated with this ODE. Each scheme should ultimately take the form Un+1 = A(At) Un Un+1 Un where A(At) is a 2 x 2 matrix to be determined. (d) For each scheme, there exists an integer p > 0 such that max |Aij (At) - Aii (At)| = 0((At ) 041). 1