Question

WE L L. ew 2 0VISUWURSU3121/WW.Apter Section 8//usersmisegaye BellectiveUseramtegekey=MUORAJM69GZ29FnHyxZR794HHcym (1 point) Euler's method for a first order IVPy a ,), V(o) is the the following algorithm. From (0.10) we define a sequence of approximations to the solution of the differential equation so that at the nth stage, we have In=In-1 +h, Wen-1th fan-1,-1). In this exercise we consider the IVP y = 1+ y with y(0) 2. This equation is first order with exact solution y tan(+ tan (2)). Use Euler's method with h 0.1 to approximate the solution of the differential equation For this example we include the slope field to give a rough idea what the shape of the solution should look like. Apply Euler's method to complete the following table: In the first two rows enter the values of and y, and in the third row use the exact solution to find the errorse l ) - Mal. A calculator or other scientific software would be handy to work these types of problem. You can always use answers given by explicit formulas which are very accurate. You need at least 4 significant digits. If your answer is marked wrong try entering a more accurate answer ONO

Answer 1

!! PLEASE VOTE WITH MANY THUMBS UP !!

1. y'=1ty? ylo)=2 , h=1=o.l. The Euler method states that IntiYnth f(xn. In) where anti= senth., we have h=0.1,20=0, Jo=2 f(x,y)= y²71 for 1= Roth = 0+0.150-1 Y = yo+0.1 f(0,2) = 2+0-1 (5) - 2.5 22=04+0-1= 0-2 y = y, to-1 f(0-4,2-5) = 2.5+0-1(1+(2-573)=3-225 X3=0.2 +0-1= 0-3 Yz = y2 +0.1 f10-2,3-225) = 3-225701(1+3.22533) 1 = 4.3650 625 YH = y3 +0.1 f(0.3, 4.3650625) = 4.3650625+4174.365062.j! -6.37043956 X,50 0.1 0.2 0.3 0.4 9= 22.5 3.225 4.3650625 ! 6.37043956 GEO 0-1276168 0.4796491611-6909818! 9.31984812 Yéco-1)= tom (otttan'a) (=ly (0-1)-Y(0-1) | Similanly other calculation.

Answer 2

Euler Method for Salmetion of Oedinary diferential equation Cruidez dilterential equation dy - Any), where y enod=yo where, h=dise of Sub-in then Ynynt + ht (antig ned Hn=knath Criuen IVP: y'=lty2 ,y(0)=2 it. Hoo, Huy)=1ty 2 and h=01 then Ho=0 , Hi-011 M2=0,2 = 0.3 y = yoth for yo) 2401(+W = 2+0.185 = 2+0.5 =2.5. the + C Ya = yith H H1191) 2.5+ on (it (2.512] is = d.540:18 [1+625] 2.5+0.187.25 = 2,5+0.725 3.225 G= Yath HU2192) = 3.295+(0.1) [1+ (1.225) 2] = 9:225 +0:19 [1710,400605 ] = 3.225+ 0.13 (11,400625) = 3.225 +1.1400625 74,8650 625 Ty = Goth Hugiya) > 4.9650695 + (0.1) [1+ (4 13650 625) 2] = 6.3704 395629 Cinem , chact deeln is Co = 1 ginos-yol = 12-210 ly = tan (ut tantas 9= | ychy, l = 1 ton Coilt fant (21] - 2.51 - | tam (0.17 1107149) - 0-57 - I tan (1207149)-2.51 18.697619 -0.58 0127619

= I tan (o. 2 + fant (2)] - Jidas) = tan(11907149) - 1.225/ 10.70 4651-31225) = 0.479653 anal = tan [oB tant (a)] 4. 36506251 - Itan (1.407149) - 4.3650625/=) 16.05605496-4,1650 6251 = 1169099246 (4=| tam (0.4+ tant (2)] – 6. 97640961 - 115.6909574-6-17043961 = 90199178 no 1 T kn 10 02 1. 0.3 0.4 J. 225 4.3651 6.3704 ss An lol 01 127619 0.479651 1,690992 9.319918 Slobe Hield (Vector field 1 Direction treld) a tool to geabhically obtain the salutions to a fiutoidee diHerential equation It Gouen dillerential equation is independent on w' then the Slope field is heel zontal Shittinuacient (iealong a horizontal line the slope does not way.