Question

Consider the initial-value problem y' = 2x - 3y + 1, y(1) = 9. The analytic solution is 1 2 74 -X + e-3(x - 1) 9 (a) Find a formula involving c and h for the local truncation error in the nth step if Euler's method is used. (b) Find a bound for the local truncation error in each step if h = 0.1 is used to approximate y(1.5). (Proceed as in Example 1 of Section 6.1.) (c) Approximate y(1.5) using h = 0.1 and h = 0.05 with Euler's method. See Problem 1 in Exercises 2.6. (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 global truncation error of Euler's method is O(h). (Round your answers to four decimal places.) Since y(1.5) = , the error for h = 0.1 is Eo. 1 = while the error for h = 0.05 is E0.05 och) we expect Eo.1/50.05 - 2. We actually have E0.1/50.05 (rounded to two decimal places). With global truncation error

Answer 1

yen. ", + 2? + 386-342 - 1) 9 .) You. Val 6x +33e -3in1) ti see graph of equation , IN xo, u dorsider Taylor expansion of function y around 9190+h)- yello) +h y'(%o) +igh?y" (70) +0(?) it cougide finite dritenere, y'(X), y Roth ) - gerol een write (i= noth] go the humenical solution is given by, ON 9.- Yo the (to, 30h for exact solution use equestions y noth) ylho)thylend tilh?y"! 80) + 0(43) so, the truction eroon (local), by Eulen methed in ITC = 4(90th) — 91 =14424"(no) + 0(?) If y is continery second deviative, the there exists a celno, toth] such that LTE llera Taruction bron]- ychoth) ay, = Vzh?y" (0) y lo). 3f SI 110,4190) + f(40,918)) fIV,4190) an uppun bord, I Global Function tool 2 OR (ell-rol) CV i in lipschita conehut off Now, we hool in ea , Nois. for y1.s), use tood, shinos no.1, o.1, y S40.1) - let Bis 21 - yllis) + 018 4'11.5) + 40-v?ruhan 4019}] Scanned with CamScanner

Please do like