Higher Engineering Mathematics

472 DIFFERENTIAL EQUATIONS

Table 49.17

n xn k 1 k 2 k 3 k 4 yn

0 1.0 4.0

1 1.2 2.0 2.1 2.09 2.182 4.418733

2 1.4 2.181267 2.263140 2.254953 2.330276 4.870324

3 1.6 2.329676 2.396708 2.390005 2.451675 5.348817

4 1.8 2.451183 2.506065 2.500577 2.551068 5.849335

5 2.0 2.550665 2.595599 2.591105 2.632444 6.367886

4.k 3 =f

(

x 1 +

h

2

,y 1 +

h

2

k 2

)

=f

(

1. 2 +

0. 2

2

,4. 418733 +

0. 2

2

(2.263140)

)

=f( 1 .3, 4. 645047 )=3(1+ 1 .3)− 4. 645047

=2.254953

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

=f(1. 2 + 0 .2, 4. 418733 + 0 .2(2.254953))

=f(1.4, 4.869724)=3(1+ 1 .4)− 4. 869724

=2.330276

6.yn+ 1 =yn+

h

6

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

n=1:

y 2 =y 1 +

h

6

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

= 4. 418733 +

0. 2

6

{ 2. 181267 +2(2.263140)

+2(2.254953)+ 2. 330276 }

= 4. 418733 +

0. 2

6

{ 13. 547729 }=4.870324

This completes the third row of Table 49.17. In a
similar mannery 3 ,y 4 andy 5 can be calculated and
the results are as shown in Table 49.17. As in the
previous problem such a table is best produced by
using aspreadsheet.
This problem is the same as problem 1, page 461
which used Euler’s method, and problem 5, page 467
which used the Euler-Cauchy method, and a compar-
ison of results can be made.

The differential equation

dy

dx

=3(1+x)−ymay

be solved analytically using the integrating factor

method of chapter 48, with the solution:

y= 3 x+e^1 −x

Substituting values ofxof 1.0, 1.2, 1.4,..., 2.0 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 49.18 on

page 473.

It is seen from Table 49.18 thatthe Runge-Kutta

method is exact, correct to 4 decimal places.

The percentage error in the Runge-Kutta method

when, say,x= 1 .6 is:

(

5. 348811636 − 5. 348817

5. 348811636

)

×100%=− 0 .0001%

From problem 6, page 468, whenx=1.6, the per-

centage error for the Euler method was 0.688%, and

for the Euler-Cauchy method−0.048%. Clearly, the

Runge-Kutta method is the most accurate of the three

methods.

Now try the following exercise.

Exercise 187 Further problems on the

Runge-Kutta method

1. Apply the Runge-Kutta method to solve

the differential equation:

dy

dx

= 3 −

y

x

for the

range 1.0(0.1)1.5, given that the initial con-

ditions thatx=1 wheny=2.

[see Table 49.19]