For the ODE dx $(

For the ODE f (x, t) With x(t0) a second-order-accurate (RK2) iterative scheme for solving initial-value ODES generates the sequence of approximate solutions to, at times tj = to-hhj from the recursion + 3k2). (a) By examining the Taylor series in h and comparing terms at O(h2), show this method is second-order accurate. (b) A damped oscillator Obeys the second-order ODE up + 27 + & = o, subject to the boundary conditions O, = I i. Describe how the shooting method would solve this ODE. ii. Find the analytic solution to this ODE. iii. Find the approximate solution generated by the RK2 method above with step- size h = r/ 2 with initial data $(0) O, = c and find the value Of c that gives an approximate solution consistent with both boundary constraints. [61