Question

[10pt] 5. Consider the IVP :' = t +x?, *(0) = 1. Complete the following table for the numerical solutions of given IVP with step-size h = 0.05. t by Euler's Method by Improved Euler's Method 0 0.05 0.1 6. Which of the followings is the solution of the IVP

Answer 1

Step-size > Given IvP is s'= t +212 x(0)=1. and h = 0.05 so If x'=f(d, 2) be differntial equation with n (to)= Ho and if h be the then Euler method States that Hati = Hn tof (thian) tntilathth n=0,1,2,ar. and the Improved. Ewer method state that Hati Xat! into = Hn + hf (2h, th] stha la tath n=0,1,2,-- Here f(tr) = +²+42 Ho = 1, to = 0, += 0.05 t = 0.1 n=0.05, NOW, By Ewer method xe (0.05) = Ny = Hot hf to, ko) 1 +(0.05).f(0,2) -+(0.05).1 - 1.05 = In thelf (th, Hla) + f(thti, Kints)}

For Improved ever method , N, -1.0526 aso, By Improved Euler method Hi = Hot Ą {f(do , Xo) + f ( to, XJ} where, 5, = Xothf (to, do) = 1.05 2,5 ] +(005) {f(0,1)+F(0.05,1-05)} I 1+ (0.025){1+ 1.105) ē 1 + 0.0526 = 1.0526 Ewer method, Hit hf (d., N) H = 1.05 Now, sind From ..x (Oil) = ₂ = 1.05+0.05) 4(0.05,1:05) 1.05 +0.05). (1103) 1.05 +0.0552 - 1.2052 .:1 (0:1) = H2 = 4, the {f (fp, Mp) + f (59)} H = Hith f ( 1 ) 2= 1.0526+0.05).(0.05, 1.0526) - 1. 1084

H2 = 1.0526 + (0.05) (0.1, 11/081 11081)} = 1.0526 + (0.025). (1.1105 + 1*2379) 1860.05, 1.0526)+8 = 1.0526+ 0.05871 EI 1031. d e by Euler method se by Improved Ewer method 1 1 1.0526 Os 05 I 1.05 1.1052 L1131