Question

(1 рon Euler's method for a first order MP y-f(x.y), y(xa) - y s the the folowing algorithm. From (x.yo) we define a sequence of approximations to the solution of the differential equation so that at the nth stage, we have x h y,- -+h f(x1--1) In this exercise we consider the NPy--x+ywith y(2) 2. This equation is first order inear with exact solution y 1 4 x- Use Euler's method with h-0.1 to approximate the solution of the diferential equation For this example we include the slope field to give a rough idea what the shape of the solution should look lke. We have also plotted the exact solution y1+x- over a small interval Apply Euer's method to complete the following table in the first two rows enter the values of X and y, and in the third row use the exact solution to find the erors e, yx.)-l A calculator or other acientifc software would be handy to work these types of problem. You can always use answers given by axpilcit formulas which are very accurate. You need at least 4 signiicant digits. If your answer is marked wrong try entering a more accurate answer 4 xa 2 y 2

Answer 1

Given Iup y=-aty, g(2=2. which has exact solution Yz1+2-12- 2 0 since 42)=2 p.e. goly say Then do=2, 4=2. from Euler's method Take y'= -aty Then qo=Douth, 9=9 th flanish) let it be ey'=f(0,4) (2,4)=-x+4. e ose huo. fus n=1 9, = Both, 9,=4ot bf (aoyo) a, = 2101, 9=2+ Con [2oto] 21 = 211, = 2+Cool) (-2+2) 7,20 9, = 2+-60.(O) = 210 = 2. * =201, 9=2. also en=lg(0)-gn -e,=1466)-9,1 = /25 (201)-2/ = /14211-221-2_zl -111-com e = 0.00 5170918 (tram () Hon=2 d2=2,th, 9=9,+ b faq y) 92 224 +0.1, 4 = 2+ Cow Gang)

22=22, %= 2+ (0-1) (2.1+2) Yz2 -0.01 1 2 =2.2 4 = 1.99 also ez=14643)-4,1 = 1962.2) - 1991 =(1+2-2-2-2-2-1-991 S = 001140 2358. for n=3 &z= 12th , 33= 9th f(22,42) $3=2021001, 43 = 081.99 + (0-1) (laty) 23 = 2.3 -1.99+Co.1) Go yz = 1.969 ez= 14(*)33) = 1y (23)_109691 -1.99+C.1) (-2,271-99) also +2.3 - 02-3-2 tos n= -1.969 es I +0:618858807 4 *4=8gth, gu = 4 + bf (334) $4=2.3+01, y = 1.969+ (0-1) (-2374) dy=2-4, S4=1.969+(0.1) (-2.3+1.969) 44 =1-9359 and eu=14 (24)-94)=14 (24) - 1.93591 = /14.24 e 24-2_1.9359) -0.027724697

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