Consider the initial-value problem y' = 2x − 3y + 1, y(1) = 7.The analytic solution is y(x) = 1/9 + 2/3 x + (56/9) e^(−3(x −1)).
(a) Find a formula involving c and h for the local truncationerror in the nth step if Euler's method is used.
(b) Find a bound for the local truncation error in each step ifh = 0.1 is used to approximate y(1.5). (Proceed as in thisexample.)
(c) Approximate y(1.5) using h = 0.1 and h = 0.05 with Euler'smethod. (Round your answers to four decimal places.)
h = 0.1         y(1.5) ≈______
h = 0.05        y(1.5)≈______
(d) Calculate the errors in part (c) and verify that the globaltruncation error of Euler's method is O(h). (Round your answers tofour decimal places.) Since y(1.5) =______, the error for h = 0.1is E_0.1 = ______, while the error for h = 0.05 is E_0.05 = ______.With global truncation error O(h) we expect E_0.1/E_0.05 ≈ 2. Weactually have E_0.1/E_0.05 = _______ (rounded to two decimalplaces).
