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i Safari File Edit View History Bookmarks Window Help 79% U. u.s. Sun 8:40 PM webwork.math.ttu.edu | WeBWorK: f17gkem3350s008 HW4:9 l Safari File Edit View History Boakmarks Window H... Chegg.com Chegg Study Guided Solutions and Study Help Chegg.com (1 point) Euler's method for a first order lVP y =f(x,y), y Xo)-yo is the the following algorithm. From (xOM) we define a sequence of approximations to the solution of the differential equation so that at the nth stage, we have 1.pdf In this exercise we consider the IVP y' =-x + y with y(0) =-1 . This equation is first order linear with exact solution y = 1 + x-2er Use Euler's method with h = 0.3 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. We have also plotted the exact solution y 1+x-2e over a small interval Apply Euler's method to complete the following table: In the first two rows enter the values ofx and y and in the third row use the exact solution to find the errorserlA calculator or other scientific software would be handy to work these types of problem. 11.3 df

Explanation / Answer

Given y'=-x+y with y(0)=-1

Using Euler's method :

h=0.3

f(x,y)=y'= -x+y

when n=0 x0 =0 with y0=-1 so, f(0,-1)=-1=f0

so, y1 = y0 +h f0 = -1 + (0.3)(-1) =-1.3 and x1 =0.3

when n=1 x1 =0.3 with y1=-1.3 so, f(0.3,-1.3)=-1.6=f1

so, y2 = y1 +h f1 = -1.3 + (0.3)(-1.6) =-1.78 and x2 =0.6

when n=2 x2 =0.6 with y2=-1.78 so, f(0.6,-1.78)=-2.38=f2

so, y3 = y2 +h f2 = -1.78 + (0.3)(-2.38) =-2.6844 and x3 =0.9

when n=3 x3 =0.9 with y3=-2.6844 so, f(0.9,-2.6844)=-3.5844=f3

so, y4 = y3 +h f3 = -2.6844 + (0.3)(-3.5844) =-3.75972 and x4 =1.2

0

n

0

1 2 3 4 xn 0 0.3 0.6 0.9 1.2 yn -1 -1.3 -1.78 -2.6844 -3.75972 error 0 0.0997 0.264238 0.3348 0.6805

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