Higher Engineering Mathematics, Sixth Edition

472 Higher Engineering Mathematics

Table 49.15

n xn k 1 k 2 k 3 k 4 yn

0 0 2

1 0.1 2.0 2.05 2.0525 2.10525 2.205171

2 0.2 2.105171 2.160430 2.163193 2.221490 2.421403

3 0.3 2.221403 2.282473 2.285527 2.349956 2.649859

4 0.4 2.349859 2.417339 2.420726 2.491932 2.891824

5 0.5 2.491824 2.566415 2.570145 2.648838 3.148720

Letn=1 to determiney 2 :

2. k 1 =f(x 1 ,y 1 )=f( 0. 1 , 2. 205171 );since
   dy
   dx

=y−x,f( 0. 1 , 2. 205171 )

= 2. 205171 − 0. 1 =2.105171

3. k 2 =f

(

x 1 +

h

2

,y 1 +

h

2

k 1

)

=f

(

0. 1 +

0. 1

2

, 2. 205171 +

0. 1

2

( 2. 105171 )

)

=f( 0. 15 , 2. 31042955 )

= 2. 31042955 − 0. 15 =2.160430

4. k 3 =f

(

x 1 +

h

2

,y 1 +

h

2

k 2

)

=f

(

0. 1 +

0. 1

2

, 2. 205171 +

0. 1

2

( 2. 160430 )

)

=f( 0. 15 , 2. 3131925 )= 2. 3131925 − 0. 15

=2.163193

5. k 4 = f(x 1 +h,y 1 +hk 3 )

= f( 0. 1 + 0. 1 , 2. 205171 + 0. 1 ( 2. 163193 ))

= f( 0. 2 , 2. 421490 )

= 2. 421490 − 0. 2 =2.221490

6. yn+ 1 =yn+

h

6

{k 1 + 2 k 2 + 2 k 3 +k 4 }

and whenn=1:

y 2 =y 1 +

h

6

{k 1 + 2 k 2 + 2 k 3 +k 4 }

= 2. 205171 +

0. 1

6

{ 2. 105171 + 2 ( 2. 160430 )

+ 2 ( 2. 163193 )+ 2. 221490 }

= 2. 205171 +

0. 1

6

{ 12. 973907 }=2.421403

This completes the thirdrow of Table 49.15. In a similar

mannery 3 ,y 4 andy 5 can becalculated and theresults are

as shown in Table 49.15. Such a table is best produced

by using aspreadsheet, such as Microsoft Excel.

This problem is the same as problem 3, page 459 which

used Euler’s method, and problem 4, page 461 which

used the improved Euler’s method, and a comparison of

results can be made.

The differential equation

dy

dx

=y−xmay be solved

analytically using the integrating factor method of

chapter 48, with the solution:

y=x+ 1 +ex

Substituting values ofxof 0, 0.1, 0.2,..., 0.5 will give

the exact values. A comparison of the results obtained

by Euler’s method, the Euler-Cauchy method and the

Runga-Kutta method, together with the exact values is

shown in Table 49.16.

It is seen from Table 49.16 that the Runge-Kutta

method is exact, correct to 5 decimal places.

Problem 8. Obtain a numerical solution of the

differential equation:

dy

dx

= 3 ( 1 +x)−yin the

range 1.0(0.2)2.0, using the Runge-Kutta

method, given the initial conditions thatx=1.0

wheny=4.0.