Consider the initial-value problem
(a) Compute estimates of x(2) using the second order Adams-Bash forth scheme (using the second-order predictor-corrector to start the computation) with step sizes h = 0.2 and 0.1. From these two estimates of x(2) estimate what step size would be needed to compute an estimate of x(2) accurate to 3dp. Compute X(2), first using your estimated step size and second using half your estimated step size. Does the required accuracy appear to have been achieved?
(b) Compute estimates of x(2) using the second order predictor-corrector scheme with step sizes h = 0.2 and 0.1. From these two estimates of x(2) estimate what step size would be needed with this scheme to compute an estimate of x(2) accurate to 3dp. Compute X(2), first using your estimated step size and second using half your estimated step size. Does the required accuracy appear to have been achieved?
(c) Compute estimates of x(2) using the fourth order Runge-Kutta scheme with step sizes h = 0.4 and 0.2. From these two estimates of x(2) estimate what step size would be needed to compute an estimate of x(2) accurate to 5 dp. Compute X(3), first using your estimated step size and second using half your estimated step size. Does the required accuracy appear to have been achieved?