Question

1. (a) Determine the solution to the IVP = 3+2-21 x(0) = 2 X = ltttle-2t Dox + 2x²0 X = 0 ? A + 2x=3+27 2x=0 X = ezt 2u - S 3+2+ = $(3+2+) e?edt (= Xw = [² (3+2+) e 2t_te 2+] = "2+ X=1 ttte-2 xw = 1+t ttte-26) (b) Evaluate your solution at t = 1.- 2= title c=0 [Xs=1+t) (c) Using Improved Euler's Method i. Compute approximate point (t1,2]) with step size h = 1. ii. Compute approximate points (t1, 2) and (12,1%) with step size h .5. (d) i. What is the error between the first IEM approximate at 1-1 and the exact solution? ii. What is the error between the second IEM approximate at t= 1 and the exact solution?

please answer c through d for a thumbs up. thank you.

Answer 1

Jos ive= f(t, 2), alto) = 20 Improve Euler's Method is : Zeits = L; +hf(ti.) 2+1 = b + a 1 f (ti, 4) + f(tity, 7:) 7 tiri = tith Here we are given, ď = 3+2t-zx x(0)=2 :: € (t, x) = 3+2t - det Xo = 2, to=0 (chi) hat = tot h = 0+1 = 1 flto, 6) = 3+ 2 to - 206 = 3+0 - 4 = -1 Q = To + hf 1o, 2o) - 2 + 1 1) Scanned with CamScanner

flte, q ) = 3+2 t, - 250 = 3 + 2x1 - 2X1 =3 Jy = 8 + h I fto, a) + f (te, 5g)] = 2 + ] [ -- +3] 아 = 2+1 - 4 =3 : (+ 1 ) + (1, 3) h =0.5 ist iteration ti= to th=0+ 0.5=0.5 = 3 + 2x0 - 2x2 = 3 - 4 = -1 r = Xoth f (to , %o) = a + (0.5)(-1) = -0.5 = 1.5 &lth, ñ) = 3+2 ti-2 ų = 3+2x0.5-2x1.5 Scanned with CamScanner

Q = Mo + ✓ Alto, 26) + flt., õ) = 8 + 0[ -1 + 1] -d + bien xo : (to, ) = (0.5,2) qituation : toe te th = 0.5 +0.5 = 1 Alth,04) = 3 + 2+1 -20, = 3 + 2x0.5 - 2x2 = 3 + 1- 4 ño = th flt.,ah) a + 0.5 x 0 = = m + 1 [f(t1, XD+ fltp, 6:37 f(ta. a) = f(212) = 3 + 2x1 - 2x2 = 1 Cs Scanned with CamScanner

| : 32 - 4 + : [ ] = 8+ lo pfo = 2.25 | ( 4, 3,) : (1 , 3 4 5) (d.) We will first given IVP. find exact solution of i = 3+at-ga a + 9x = 3+ at . It . Integrating factor , 1.f. = e Jode edt 2. solution is: & lt) 1.F. - S (3+2t) I. F. de A x (t) pat - J (3+2+) eat at = fzeht dt + (at eht at Scanned with CamScanner

consider Steht at = + Seat at - S2410) Secl ct} de Eternet - Succeed at X + Sedt +2 / teat + c implies alt) étt - Bedt tolten - Cee)+C ( m integration constad) :: 21t) eart - Belt + edt (+ - $) + c * all) este = edt [+ + 3 -1]+ 1 xH) = (t +1) + ce et Now use x (0) = 2 to find C. 210) = (0+1) + ceo 2 = ceº +1 a FIEC] (as e' = 1) cs Scanned with CamScanner

& Solution in X(t) = (t +1 +1.0 2t ifa(t) = 7+1+ edt . (L) = 1 + 1 + 7-21 x(i) = 2.135335 283 → exact solution at tal () solution error between the first IEM approximate at t=2 and escact in «(L) – - 12.135335 283 - 3 = 0.864664717 li) error between the second IEM at t=1 and exact solution in a I accio - xq1 = 12.135335483 - 2.251 0.11 4664717 Scanned with CamScanner