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Chapter 3: Q 3.6-8E (page 130)

Use the improved Euler鈥檚 method subroutine with step size h= 0.2 to approximate the solution to the initial value problemy'=1x(y2+y),y(1)=1 at the points x= 1.2, 1.4, 1.6, and 1.8. (Thus, input N= 4.) Compare these approximations with those obtained using Euler鈥檚 method (see Exercises 1.4, Problem 6, page 28).

Short Answer

Expert verified

xn

yn

1.2

1.48

1.4

2.24788

1.6

3.6518

1.8

6.88733

Step by step solution

01

Find the equation of approximation value

Here, y'=1x(y2+y),y(1)=0 for 1猢�x猢�1.8

For h=0.2, x=1, y=1, N=4

F=f(x,y)=1x(y2+y)G=f(x+h,y+hF)=1x+0.2y+0.2x(y2+y)2+y+0.2x(y2+y)

02

Solve for x1 and y1

Apply initial points xo=1,yo=1,h=0.2

F(1,1)=2G(1,1)=2.8

x1=1+0.2=1.2y1=1+0.22(2+2.8)=1.48

03

Evaluate the value of  x2 and y2

F(1.2,1.48)=3.05867G(1.2,1.48)=4.61934

x2=1.2+0.2=1.4y2=1.48+0.1(3.05867+4.61934)=2.24788

04

Determine the value of  x3 and y3

F(1.4,2.24788)=5.21489G(1.4,2.2488)=8.82538

x3=1.4+0.2=1.6y3=2.24788+0.1(5.21489+8.82538)=3.6518

05

Determine the value of  x4 and y4

F(1.6,3.6518)=10.6172G(1.6,3.6518)=21.7381

x4=1.6+0.2=1.8y4=3.6518+0.1(10.6172+21.7381)=6.88733

Hence the solution is

xn

yn

1.2

1.48

1.4

2.24788

1.6

3.6518

1.8

6.88733

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