9.6 Ill-conditioned and sensitive ODEs (a). Consider the ODE x = 10x 11t 5t2 1, x(0) = . Show that the exact solution is: x(t) = e10t t2/2 t. Now, set = 0. Solve the ODE on the time interval t = [0, 3] with your RK4 method with h = 28. Plot the numerical solution and the exact solution on 0 t 3. What do you observe? What do you think accounts for the discrepancy between the numerical and exact solution? (b). Dahlquist and Bj ork are credited for this pathological example. Consider the ODE x(t) = 100(sin t x), x(0) = 0. Solve it with your RK4 method on 0 t 3, with h = 0.015, 0.020, 0.025, 0.030. Plot all these solutions. Observe the numerical instability