Question

. Consider the IVP y'= 1 + y?, y(0) = 0 a. Solve the IVP analytically b. Using step size 0.1, approximate y(0.5) using Euler's Method c. Using step size 0.1, approximate y(0.5) using Euler's Improved Method d. Find the error between the analytic solution and both methods at each step

Answer 1

son:- Griven IVP .dy dhe y' = i+y? y (0) = 0 Ity 2 =/da In [Taking integral on both sides oy Tty or tant(y) to Ćis a constant By the initial condition y20 fan (0) ota Then solution of ☺ is tenty y = tense ny y'= 6 Giren differential equation = 1+ y2 yo)=0. =flrn) , (sag) Here f(x,y) = 1 + y2 , No=0, y =0, h=onl Now the iteration formula for Euler's method h In th flunt, Int) put راد ۱ m 2 Yoth fatto, Y. ) 0 + oil f(0,0) 0 + (0.1) . 1 0.1

putting nz2 Ye=Yith fe M, Yil **1 +(01) 6+ (y 0.101 0 201 put th f(x2, 72) 3 = 0+201 +0.0) (1 +0.2017 0 305 C correct upto 3 decimal places) put n=4 Yy = Ysthf (Ang ty) 0-305 + Co. (1+ (1+ (0-205)") 0.4.14 put n=5, Ys = Yuth fina, Yu) - olyan + (0.1) "(It olyong?) 0.531 Here the the sohtion of the given Ive glois) (up to sth iteration) is ar53 1 by Euler's methad. Now use Euler's improved method No=07.20 haoil Here fexrg = ty?

y (1) - Toth tch, Y.) of long (to) Now using iteration formula 6) = % + ₂ [ f (xo, Yo) + f(x, y)] Y, we get, r=2, 4 = 26th 0 + 4 + 5 [fex-9.) + f(34,5,5")] (01[ { + 5) + f(oils ] and [ + ((t (ory") 1005 ra3 7 (3) - Tot 1 2 [ freo, tu) + H x,y)] 1+ (1+ 0.1805²) - ot 2 01005 2 24 (") - Yot y [ F4, Yo) + f (6, 7, ®)] rog2) 1 + 2 (1+ orlous- Hence the of the IVP by solution given Improved method is Euler's Y(05) - 0-1005- firs)