Question
Solve the following problem over the interval from x = 0 to x = 4 using a step size of 0.25 where y (0) = 2.
Homework Answers
Answer #1
Page No: Date 1 0. 88 dy - 40 - 0.54 14 (0)=2 Step size h = 0.25 @ Euler's method. 4 - yot hf (xoiyo) (Xn+ 1 = Xooth) - 2+ (0.25) f (0.2) = 2 +10.75) = 2.75 Y = 2.75. [xo = o) Y₂ - 4,+ hf (x, y ) = 2.75 + hf 2o.25, 2.75) = 2.75+ (0.25) (3.51061) = 3.62765. 42 = 3.627 65 [x = xoth = 0.25). on the same way we find other approximation as followed Y = 4.66602 X2=81+ h = 0.5 Y q = 5.90489 x3=X2+ h = 0.75 x 4 = x3 + b = 1 Yo = 7.39232 Y6 = 9.18656 25 = X4+ 6 = 1.25 Y7 = 11.35836 xo=x5th = 1.5 Y8 = 13.99376 X7 = X5th = 1.75 Yg = 17.19752 92 = 21.09752 Rg=x8th = 2.25 Y = 25.84939 X10 = Xg+h = 2.5 - X11 = 210th= 2.75 Y 13 = 38.71 ; X12=X11th = 3 914 47 33587 ; 913 = X12+ 6 = 3.25 Y 5 = 57.86353 X 14 = 213 + b = 3.5 16 = 70.71612 x 15 = xigth=3.75 Klo=2,15 h = 4. y (4) = 70.71612 kg = X7th - 2 912 3164323 20
SALURU utive E MP, Flos Belalu D/03.20. R SHON Ke-Procure SWD. Mahi missioner. B mentioned Page No: Date / Heun's method (9mproved Euler's metha (0.8x dy - geo - 0.54 (0)=2 dx h = 0.254 (4)=2 t the improved Euler's method states that Into Y + h ( f (x no Yn) + t ( Rent 9 7 Ynt Yuth of (xn. Yn) Hork 2 L delling of ary and Drains. & Allied sich are various evapura and Xnt = xnth. ale have. - h = 0.25 = /4 x0=0 190=2. f(x,y)=-4 + 4 + 5 2 (0.8)x. - -054 +4 e By step Step 1: x1= xoth - 0 + 1 - 1 Y = yoth & f (80, 4o) = 2+ hof (012) = 2 + :3 = 1 y = Yo + h. [f(xo, Yo + f (x 32+ 5 f (0.2) +(+-+)] - 2 + (3+4 02-10-5) Stef = 2.813826379. step. 20 x2= Xith = 1 Y₂ = 4, + h f(x, 14, ) =
love story and change vows the Jetstar flight 9201. travelling between their home cities Page No: Date 4 = 2.813826+ f . 2-8 13 826) D ? = (2.813820 +1 -1.40 6.91382 +4 es). Page No: - e's +2.462098081 2 l 42 = 9+ h [ f (x, y) + f(x,yen! - 2.81382 + B | (2.81382) + F(4 10'5+2.46209) = 3. 764357) By Above process we find other steps. as follows step . X = X₂ + h = 1 + 1 = 3/4. ŷ - ₂ + h f (x2 , 42 ) = €245 +3.29 38125. Y3 = 92 + 2 f (x 2, G2) + f (X31 Ý3) = 4.88695425 Step.co x4=X3th = 3/4+ 1 = 1 y = y2 + b f (X3143) =é +4.276084. Y = Y3 + h (f (8314₃) + f(x4, 94 ) Ya = 6-22421174
Date SALURU Step 5: X5 stepe SHO e-Proeun use 5 +5.446185 ssione B oned Step 95 = 7-827627 Step 6. x6 = 32 Yo = 9s + h f(X5, 45) =e+ 6.849 1736 y and Step Y6 = 95 + h (f(x5, 45) + f(xo, 967 = 9.75 9633 Step 7 ste x7= 7/4 G = 45+ 8.539679. Y = 12.0960777 Ste Step. 8: x8 = 2. ģe - es+10:58406806 Y8 = 14 9 29 234 Step.gr xg = 9/14 To 93rés +13.063 080 Yg = 18.37 1490 Step-10 x = 5/2 9 10 = 5 + 16.07505 4 10 = 22.559830
Page No: Date u Steb 211 = 24 ģv = +19.7398515 911 = 27.66 131901) Step 12 ! x 12 = 3 9 12 = e '5 +24. 203 654 13 912 = 33.879789 XA = 7/2 jie = ls + 36.2810092 914 = 50.7176556 Step 14 Step 15 215 = 15/4 915 = 5 + 44.377948. 915 = 6200 1148 1791 Step-16 4 x 16 = - 3 è + 54.26004656. 916 = 75.79819877 9(4) = 75.798198 71.
Page No: Date (c) mid Point method: (x= SALURU -- B tive PMP 9 Flod Ber ENDOS 2013 OR SHORT Fok Procurer per SWD, Maha mmissioner, BB w mentioned y = 4 0.8 -0.5.4. 4 (0)=2 h = 0.25 7(4) = ? Work Sol! modelling of ndary and ater Drains S & Allied which are various adevapura K, - hf (xoiya) = (0.25). f (012) = = 0.25).(3) - 0.75 K2= hf / not h, yo t ) = 70.25) F (0.125, 2.375) = 0.8083 y = yo tk2=2+0.8083 y = 2.8083 g10.25) = 2-8083 again taking (x,y) in Place of (Royo Ri= h f(x,y) = 0.87037 Kz- hf (x + h y itki - 0.94442. Y₂ = 4 + K2= 3.75272. y (0-5) = 3.75272 again taking (x2142) in Plance of Cxarye) Kiz h F(X2142) = 1.02273 K2= hf (x2th, 92 + k 1) = 1.1157) 43 = 42 +K2 4.86843. yto 75) = 4.86843
Page No: Date / (X343) (xogo) 8. Ki= 1.21357 K2= 1.32935 Y = 4 3 +K 2 = 6.19778 y (1)=6.19778 = (x+, +) = (Xoryo) Ri= T: 45082 K2= 1.5942 95 = 9+ + K2= 7.79199 y (125) = 7.79199 = (X5, 45) = (xo iyot Ri= 1.74428 K2 = 1.92115 Yo = 9.71314 401,5) = 9.713.14 (xo. Yo =) (8019) KI = 2.10597 K2= 2.32 353 4 - 12.03667 9.01.75) = 12.03667 » (X7,97) =) (Xo1yo) KI = 2.55062 K2= 2.81769 Y8 = 14.85436 (2) = 14.85436
Page No: COCO Date / ngalu - WGALURI ecutive BA9th Fid END OS 19- OR SHRI Bok e-Procu per SWD. Man missioner. Bi ow mentioned = (xg, Y8) = (xo, you Ki= 3.09624 K2= 3.42 364 Yg= 18.278 9(2.25) = 18.278 Work hodelling of ndary and fer Drains. is & Allied which are various devapura (xg, Yg) = (xo 198) KI= 3.7649 K2= 4.16584 4 10 = 22.44 384 4(2-5) = 22.44385 > Wo, 910) =) (X0190) Ki= 4.58 358 K2=5.07422 Y = 27.5 18 05 y (275)=27.5 1805 - lan, y) = (Xoyo) R = 5.58526 K2 = 6.18 535 9,2 = 33.70 34 y (3) = 33.7034 (X12 14 12) =) ( X014) KI= 6.81025 K2 = 7.54393 913 = 41.24 733 y (3.25) 41.24733
Page No: Date / (813 413) = (xoiyo) Kiz 8.30 782 K₂=9.20458 y 14 = 50.4519 (3.5)= 50.4519 (X14, Y 4) = (xolo) Ki= 10.13816 K2 = 11.23402 915= 61. 68593 91375)= 61:08593 - (x15, 9,5) = (Xolyo) KI= 1203748 K2 = 13.71379 9,6 = 75.3997) (4 (4)= 75.3997)
epast am and Tomor Page No: Date 1 HANA er, SW Bloch 50 0111 a RM 1 tal of Trond ites Runga Kutta 4th order method y = 4 6 8x 0.54 9(0)=2., h = 4 y (4)= ? Ki= b f(xol yo no = 4 f (012)= 4.3 = 12 ut to ка: ** ( xo +Че+ ка) = 4 f/2,83.6742875 = 4.(15.81213) + f(218) - = 63.24852 K3= hf (xot h , got ke = 4F (2,33.62426) - 4.3 = 12 . K4 = hf (xoth, yot K3) = 4€14,14) = 4.194.13012) =_364.5208 Yi - Yo+ I (Ki+2K2+2 K3 + Kg) 9,= 27+(12+2163-24 852) +2012) +(364.5200)| Y,= 89.83625 7(4) = 89.8362.5.
Answer #2
Page No: Date 1 0. 88 dy - 40 - 0.54 14 (0)=2 Step size h = 0.25 @ Euler's method. 4 - yot hf (xoiyo) (Xn+ 1 = Xooth) - 2+ (0.25) f (0.2) = 2 +10.75) = 2.75 Y = 2.75. [xo = o) Y₂ - 4,+ hf (x, y ) = 2.75 + hf 2o.25, 2.75) = 2.75+ (0.25) (3.51061) = 3.62765. 42 = 3.627 65 [x = xoth = 0.25). on the same way we find other approximation as followed Y = 4.66602 X2=81+ h = 0.5 Y q = 5.90489 x3=X2+ h = 0.75 x 4 = x3 + b = 1 Yo = 7.39232 Y6 = 9.18656 25 = X4+ 6 = 1.25 Y7 = 11.35836 xo=x5th = 1.5 Y8 = 13.99376 X7 = X5th = 1.75 Yg = 17.19752 92 = 21.09752 Rg=x8th = 2.25 Y = 25.84939 X10 = Xg+h = 2.5 - X11 = 210th= 2.75 Y 13 = 38.71 ; X12=X11th = 3 914 47 33587 ; 913 = X12+ 6 = 3.25 Y 5 = 57.86353 X 14 = 213 + b = 3.5 16 = 70.71612 x 15 = xigth=3.75 Klo=2,15 h = 4. y (4) = 70.71612 kg = X7th - 2 912 3164323 20
SALURU utive E MP, Flos Belalu D/03.20. R SHON Ke-Procure SWD. Mahi missioner. B mentioned Page No: Date / Heun's method (9mproved Euler's metha (0.8x dy - geo - 0.54 (0)=2 dx h = 0.254 (4)=2 t the improved Euler's method states that Into Y + h ( f (x no Yn) + t ( Rent 9 7 Ynt Yuth of (xn. Yn) Hork 2 L delling of ary and Drains. & Allied sich are various evapura and Xnt = xnth. ale have. - h = 0.25 = /4 x0=0 190=2. f(x,y)=-4 + 4 + 5 2 (0.8)x. - -054 +4 e By step Step 1: x1= xoth - 0 + 1 - 1 Y = yoth & f (80, 4o) = 2+ hof (012) = 2 + :3 = 1 y = Yo + h. [f(xo, Yo + f (x 32+ 5 f (0.2) +(+-+)] - 2 + (3+4 02-10-5) Stef = 2.813826379. step. 20 x2= Xith = 1 Y₂ = 4, + h f(x, 14, ) =
love story and change vows the Jetstar flight 9201. travelling between their home cities Page No: Date 4 = 2.813826+ f . 2-8 13 826) D ? = (2.813820 +1 -1.40 6.91382 +4 es). Page No: - e's +2.462098081 2 l 42 = 9+ h [ f (x, y) + f(x,yen! - 2.81382 + B | (2.81382) + F(4 10'5+2.46209) = 3. 764357) By Above process we find other steps. as follows step . X = X₂ + h = 1 + 1 = 3/4. ŷ - ₂ + h f (x2 , 42 ) = €245 +3.29 38125. Y3 = 92 + 2 f (x 2, G2) + f (X31 Ý3) = 4.88695425 Step.co x4=X3th = 3/4+ 1 = 1 y = y2 + b f (X3143) =é +4.276084. Y = Y3 + h (f (8314₃) + f(x4, 94 ) Ya = 6-22421174
Date SALURU Step 5: X5 stepe SHO e-Proeun use 5 +5.446185 ssione B oned Step 95 = 7-827627 Step 6. x6 = 32 Yo = 9s + h f(X5, 45) =e+ 6.849 1736 y and Step Y6 = 95 + h (f(x5, 45) + f(xo, 967 = 9.75 9633 Step 7 ste x7= 7/4 G = 45+ 8.539679. Y = 12.0960777 Ste Step. 8: x8 = 2. ģe - es+10:58406806 Y8 = 14 9 29 234 Step.gr xg = 9/14 To 93rés +13.063 080 Yg = 18.37 1490 Step-10 x = 5/2 9 10 = 5 + 16.07505 4 10 = 22.559830
Page No: Date u Steb 211 = 24 ģv = +19.7398515 911 = 27.66 131901) Step 12 ! x 12 = 3 9 12 = e '5 +24. 203 654 13 912 = 33.879789 XA = 7/2 jie = ls + 36.2810092 914 = 50.7176556 Step 14 Step 15 215 = 15/4 915 = 5 + 44.377948. 915 = 6200 1148 1791 Step-16 4 x 16 = - 3 è + 54.26004656. 916 = 75.79819877 9(4) = 75.798198 71.
Page No: Date (c) mid Point method: (x= SALURU -- B tive PMP 9 Flod Ber ENDOS 2013 OR SHORT Fok Procurer per SWD, Maha mmissioner, BB w mentioned y = 4 0.8 -0.5.4. 4 (0)=2 h = 0.25 7(4) = ? Work Sol! modelling of ndary and ater Drains S & Allied which are various adevapura K, - hf (xoiya) = (0.25). f (012) = = 0.25).(3) - 0.75 K2= hf / not h, yo t ) = 70.25) F (0.125, 2.375) = 0.8083 y = yo tk2=2+0.8083 y = 2.8083 g10.25) = 2-8083 again taking (x,y) in Place of (Royo Ri= h f(x,y) = 0.87037 Kz- hf (x + h y itki - 0.94442. Y₂ = 4 + K2= 3.75272. y (0-5) = 3.75272 again taking (x2142) in Plance of Cxarye) Kiz h F(X2142) = 1.02273 K2= hf (x2th, 92 + k 1) = 1.1157) 43 = 42 +K2 4.86843. yto 75) = 4.86843
Page No: Date / (X343) (xogo) 8. Ki= 1.21357 K2= 1.32935 Y = 4 3 +K 2 = 6.19778 y (1)=6.19778 = (x+, +) = (Xoryo) Ri= T: 45082 K2= 1.5942 95 = 9+ + K2= 7.79199 y (125) = 7.79199 = (X5, 45) = (xo iyot Ri= 1.74428 K2 = 1.92115 Yo = 9.71314 401,5) = 9.713.14 (xo. Yo =) (8019) KI = 2.10597 K2= 2.32 353 4 - 12.03667 9.01.75) = 12.03667 » (X7,97) =) (Xo1yo) KI = 2.55062 K2= 2.81769 Y8 = 14.85436 (2) = 14.85436
Page No: COCO Date / ngalu - WGALURI ecutive BA9th Fid END OS 19- OR SHRI Bok e-Procu per SWD. Man missioner. Bi ow mentioned = (xg, Y8) = (xo, you Ki= 3.09624 K2= 3.42 364 Yg= 18.278 9(2.25) = 18.278 Work hodelling of ndary and fer Drains. is & Allied which are various devapura (xg, Yg) = (xo 198) KI= 3.7649 K2= 4.16584 4 10 = 22.44 384 4(2-5) = 22.44385 > Wo, 910) =) (X0190) Ki= 4.58 358 K2=5.07422 Y = 27.5 18 05 y (275)=27.5 1805 - lan, y) = (Xoyo) R = 5.58526 K2 = 6.18 535 9,2 = 33.70 34 y (3) = 33.7034 (X12 14 12) =) ( X014) KI= 6.81025 K2 = 7.54393 913 = 41.24 733 y (3.25) 41.24733
Page No: Date / (813 413) = (xoiyo) Kiz 8.30 782 K₂=9.20458 y 14 = 50.4519 (3.5)= 50.4519 (X14, Y 4) = (xolo) Ki= 10.13816 K2 = 11.23402 915= 61. 68593 91375)= 61:08593 - (x15, 9,5) = (Xolyo) KI= 1203748 K2 = 13.71379 9,6 = 75.3997) (4 (4)= 75.3997)
epast am and Tomor Page No: Date 1 HANA er, SW Bloch 50 0111 a RM 1 tal of Trond ites Runga Kutta 4th order method y = 4 6 8x 0.54 9(0)=2., h = 4 y (4)= ? Ki= b f(xol yo no = 4 f (012)= 4.3 = 12 ut to ка: ** ( xo +Че+ ка) = 4 f/2,83.6742875 = 4.(15.81213) + f(218) - = 63.24852 K3= hf (xot h , got ke = 4F (2,33.62426) - 4.3 = 12 . K4 = hf (xoth, yot K3) = 4€14,14) = 4.194.13012) =_364.5208 Yi - Yo+ I (Ki+2K2+2 K3 + Kg) 9,= 27+(12+2163-24 852) +2012) +(364.5200)| Y,= 89.83625 7(4) = 89.8362.5.
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