Question

1. Consider the IVP y = 1 - 100(y-t), y(0) = 0.5. (a) Find the exact solution. (b) Use the Forward Euler, Heun, and Backward Euler methods to find approximate solu- tions ont € 0, 0.5], using h = 0.25. Plot all four solutions (exact and three approxima- tions) on the same graph. (c) Maple's approximation is plotted, along with the direction field, in Figure 1. Use it, and the exact solution, to explain the behaviours observed in your numerical approximations.

Answer 1

Here yo = 0.5 to=o. I cal y'= 1-100ly 11-t) / dy = 1-100y + loot dt dy + 1007 = 1+100t this is what equat.com at integrating facter - er loodt eloot y eloot = | citloot) elook at ye loot = (toot) e loot - plook to now, using y 101=0.5, 100 100 100 00 soi exact clution becomes, ye loot = (t loot) e loot - élook +0.5 100 100 y = 1 troot -1 +0.5x100 e loot +0.5x100e loot 100 y = loot t to.re loot ly = suis is the exact solution.

(6) Forwald aulee Method h = 0.25 ttto 0.5], tn=toanh to=0 t = tot 0.25 +0.25 +2 = 0.5 fltry) = 1-1oolyt) Forward Enter's method is, yot - In the bla, ya). — Я— 86 + + ( 1 1 4 ) = 0.5+ 0.258 1-1001Yo-toly = 0.5 +0.25 f 1-100(0.5)} = 0.5 +0.25 { 1 - 501 = -11.75 7 42 = Yeth fltr, yi) = -11.45 + 0.25 { 1-1001-11.45 - 0.25) 4 = 28815 43 = Yathfidz 142) = 288.5 + 02 (1-100 (288-5--05) } Валкираеd. Cut 4 4 лооd Ynt a Yathf (Yatient) - y = Yoth flts, yel J = 0.5+ 0.25 l 1-1001 y, -0.251) y = 0.5+ 0.25 (1-1004, +25) 2641 = + y = 0.26 923

Y2 = Yg thf (d 2 1/2) Y2 = 0.26923 +0.254 1-100L Y 2-0.519 Y2 = 0.26923 10.25 { 1 +50-10042 } 72 - 13.01923 - 2542 Iz = 0·50074 HEON Method in+t -, - hССА». ) + {1, т.1, that, , . 12) = y 1 = yo + 0.25 { f ( to, yol + f (to 10:25, yo 10:25 f ( to yo)} = 0.5 +0.28{-49 + 1-100 1 yo +0.25 x (-49) - to-0-27-)} 01570.25 { -49 + 1-10010-5+0.25% (-49) - 0.25-)) = 144.5 - Y2 = y + 0.25 { 1-100 (144.5-0.25) + 1-100 ly,the itigil-(lith) = 14415 +0.25 {I-lool14415-0125) + 1-100l 144.5 +0.25{t-100 140.5- -0.25 -0.153 = 43344,75 Plotting = 0.25 Tilexact toise 100xo. 25 - 0.25 (gilfe = -11.75 (41) BE = 0.26923 yi) HM = 144.5

(Y2) exact = 0.5 10.5 100x0.5 (Y2 FE = 28815 (Y2) BE = 0.50074 (y2lum = 433 44.75 from results, we see that Backwald auler's method best approxunates yet. Texact Hem Hetted Foucard aules - Fodtward auler 0.5 10.5"