Question

YOUR TEACHER Consider the initial-value problem y = (x + y - 1)?.Y(0) - 2. Use the improved Euler's method with h = 0.1 and h = 0.05 to obtain approximate values of the solution at x = 0.5. At each step compare the approximate value with the actual value of the analytic solution. (Round your answers to four decimal places.) 0.1 y(0.5) h 0.05 (0.5) actual value Y(0.5) - Need Help? Tuto Tutor

Answer 1

Given mat fla,y) = (x+7-13, 7101= 2. h=01, h=0.05 at = 0.5 &n=nothh *h = Hot nh Q=otno. no oslo 20 en son and otno.o5 n 200 Il 0:05 5 = 40 = 0: *2 = 0.2 ² = 0.3 Improved Euler method (for h = 0.1) from, yn) + f(anth, Ynth from yon) in th)) Yoner = Youth Gi) 2 = a + 0-1 2 2+0.1 x [ n=0, = Lot Bh [f[mo, you) + f(not oily do +ot f (mo, 4) [f(0,2)+F(0.1,27 ex f(0, 2))] = 2 + 0 [(0+2-1+ f (0:1, 2:001* to+24)? ] at brade [ !+ f (0:1, 2+ 0.1 x 1)] 1+ f (0.1, 2.1)] + one [it(0.1 +211-02) 11 11 2+ o. 2+0.1 [ ec 2.122 at 01 n=l, JA (0-2) = + [ f(*.8.) + f(*th, on the f("%] = 2.122 + [ f(0.1,2.122) + f (0:1+0.1, 2.122 + 0.1 f (61,2-1)] = 2.122 + [ f(0:1, 2:122) + $(0-2, 2.122 +0.1 X f10:1,2-1227 = 2-122 +0:1 [ 1.49328 + F(0.2,0.15)) = 2 = 2.14313

n=2, = 0.2 + oil, d.14313 + Y3 10.3) = 4 + [ f(dy, My ) + f (ath I thx flag, y) 2-14313 + [ f(0:2, :14313) + f(: 0.1% f(0.2, 2.14313) ] = 2.14313 + 0[ f(0-2,2-14313) + f(0:3, 2:32352)] 4109) = 238520 n=3 8420.4) = Ya + [f43.49)+ f (*th Ys thifra,,)] = 236512 + Out [f(0:3,2136512) + f (0.3+0:1, 2:36512 + 0.1% f (0-3_436512))] =2-36512 + 0.1 [ f(03,8136512) + f(014, d.64238)] 9410.4) = 2.71231 n=4, 4510-5) = 24 + Tas [ f(^4,2x) + f (94+h, da te f (one yoj)] 75 (0-5) = 2.71231+ go [ f(0.4,2.71231) + f (0.4+0.1, 2.71231 +0.1 x F (0.4,4.71231)] = 2.71231 + (f 10.4,2.91231)+ f (0.5, 292387) Ys = 3.28878 at n=0.5

Amalytical solution Let x+y = ze I + dy Tre : du an Then, du da ) (20-12 du an :lt 202+1-gu du TI 2 dat 2+12 du 1 +(2-12 edu. tomt (60-1)) = x+c 1 9 A+ -1 = tem(+) wa tom (arc) use yo)=2 a to-1 T toml0+6) 1 = teme - C = 45 x+y=tom (x+45°) Y = 1 -x + tom ( x+45°) is analytical solution Now 310.5) = 1-0.5+ kom 10.5+45) -1-0.5 +3.40822 Y(0.5) 2.90822 Note: (4) 2 (4) are affronimake & malytical Seluhom, for second part do or repeat Seme process except h. Jake h=0.05 in each place of the Solution of first part #