Consider the following initial value problem [10 marks] y = f(t, y), f(t, y) = cos(exp(ty)) 1, y(0) = 0, (1) where 0 is a parameter. a Show that, for t [0, 1], the solution to the initial value problem (1) is confined to the region R = {(t, y)|t [0, 1], y [2, 0]}. [2 marks] b Give a formula for the absolute value of the local error |len| of Euler's method at timestep n in terms of the solution y(t) (assuming this is twice continuously differentiable) and the length of the timestep h. Show that, for the given equation (1), the solution satisfies |y (t)| 4, t [0, 1], and hence provide a bound for the local error. [3 marks] c Show that for (t, y) R, the Lipschitz constant of f with respect to the second variable satisfies L . Thus show, using the convergence theorem for Euler's method, that the global error for Euler's method applied to (1) on the interval [0, 1] satisfies |ge| 2e h, where h is the stepsize. [3 marks] d Suppose = 0 in (1). What is the order of Euler's method in this case? Justify your answer and explain why it does not contradict the results of part (c)