Question

Consider the IVP y" - 4y' + 4y = 0, y = -2, y'(0) = 1 a. Solve the IVP analytically b. Using step size 0.1, approximate y(0.5) using Euler's Method c. Find the error between the analytic solution and the approximate solution at each step

Answer 1

– we've given D£ + y^4 44+48=0 with 960) = -2; y'(0) = 1 (a) Analytically... Given De is homogeneous 2X 20 So auxillary ean mcan be wri Hen as + DP 48 +4= 0 + (0-232-0 * D = 2,2 :: Sonis + y(x) =(CI+ Cox) és : 410) = -2 = Ci – 6=-2 *Y(2) = (-2+CX) e 2x y'(x) = C₂ e 2X + (-2+ C22 ) 2 e yllo) = 1 = 62.-4.5) 12 = 5 .. 1902) = (-2+5%) e 790.5) = 1.35914091 Eulen method! let yl= 7 79110)=2(0)=1 & 7 Z-47 +4y = 0 ; 960)=-2, g'lol=1 - we're two system gegny y' = 7:= f(469417) ; 2101=1 & Z' = 47-49 = 96x79,12); Zlo)=1, 410)=-2, y'201=) Euleis method is + dy - F(, ) thens dx 9 = In th Flamin) Y = Z If

0.5 ho So for dy वर =Z= f1769, 7) Intl = Ynt hflinghiton) & d 2 = 42-44 = g (X,Y,Z) dr Lonto = Zrt. h J (Xh Inita) with Ylo)=-2go)210)= } & hort 0,1 912 pez pot n=o! Y = Yoth f(xo, To 12o) -2+0.1510.-2,1) 9,--.9 Zoy = both flrordo izo) = y +0.4 g(01-214) Zu = 2:2 In=1 Y 2 = Yith f (x1, 9, 121) --4.970-1 f10.1,7-9,2-2) → 92 = -3.68 Z2 = 24+th 9091921) = 2+2+0.1 g 101,-1.9,2:2) → 22- 3.84) 93 = y2 +hf (X2032129) --7-68+0.1f (0-2,4-683-89 73=44296 23 = 3.84 +0.1960-2,-468,3484) 163 = 6:098 ne3 44 = 11:296 +ouf f(0.3,47.29696-098) 49. =-0.6912 N=2 ! 124 = 8.98447

Forn=t. Yg = 0.20799 :: 410.5) = 0.20749 Byeuler metlog 3 Aralytically '9(0-5)=l-3591909) EC), error = Actual value-approximate value. emorú t. 1517009
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