Problem 3:

Problem 3: (Euler's method) In class, we applied Euler's method with step size h to the IVP y' = f(t, y) = t - y, y(0) =1 and found the scheme y ( tn+ 1) = y(tn) + h(tn - y(tn)). We also learned about the modified Euler method, which evaluates the derivative at the midpoint of the interval. One of you asked whether the modified Euler method is equivalent to taking two steps of the simple Euler method with half the step size. In this exercise, you will prove that this is not the case. 1. Write down the modified Euler method for the above equation with step size h while writing tn = to + nh. 2. Write down the simple Euler method for the above equation with step size h/2 and use index m instead of n, i.e., write tm = mh/2. Note that going forward two steps with step size h/2 is the same as going forward one step with step size h. 3. Write down the simple Euler method with step size h/2 for y(tm+2) in terms of y(tm+1) and use the expression that you found earlier to write y(tm+1) in terms of y(tm). 4. Now compare the modified Euler method with two steps of the simple Euler method, i.e., compare y(tn+1) from 1. with y(tm+1) from 3