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Chapter 3: Q 3.6-9E (page 130)

Use the improved Euler’s method subroutine with step size h = 0.2 to approximate the solution toat the points x = 0, 0.2, 0.4, …., 2.0. Use your answers to make a rough sketch of the solution on [0, 2].

Short Answer

Expert verified

xn

yn

0.2

0.617843

0.4

1.238642

0.6

1.736531

0.8

1.981106

1.0

1.997052

1.2

1.884609

1.4

1.724472

1.6

1.561836

1.8

1.417318

2.0

1.297794

Step by step solution

01

Find the equation of approximation value

Here , fy'=x+3cos(xy),y(0)=0or0⩽x⩽2

For h=0.2, x=0, y=0, N=10

F=f(x,y)=x+3cos(xy)G=f(x+h,y+hF)=x+0.2+3cos((x+0.2))(y+0.2(x+3cos(xy)))

02

Solve for x1 and y1

Apply initial pointsxo=0,yo=0,h=0.2

F(0,0)=3G(0,0)=3.178426

xn+1=(xn+h)yn+1=xn+h2(F+G)x1=0+0.2=0.2y1=0.617843

03

Evaluate the value of x2 and y2

F(0.2,0.617843)=3.177125G(0.2,0.617843)=3.030865x2=0.2+0.2=0.4y2=1.238642

04

Determine the all other values

Apply the same procedure for all other values and the values are

x=0.6,y=1.736531x=0.8,y=1.981106x=1,y=1.997052x=1.2,y=1.884609x=1.4,y=1.724472x=1.6,y=1.561836x=1.8,y=1.417318x=2,y=1.297794

05

Plot a graph

Hence the solution is

xn

yn

0.2

0.617843

0.4

1.238642

0.6

1.736531

0.8

1.981106

1.0

1.997052

1.2

1.884609

1.4

1.724472

1.6

1.561836

1.8

1.417318

2.0

1.297794

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