differential equation

470 DIFFERENTIAL EQUATIONS
6. y n+1 = y n + h 6 {k 1 + 2k 2 + 2k 3 + k 4 } and when
n = 0:
y 1 = y 0 + h 6 {k 1 + 2k 2 + 2k 3 + k 4 }
= 2 + 0.1 {2 + 2(2.05) + 2(2.0525)
6
+ 2.10525}
= 2 + 0.1 {12.31025} =2.205171
6
A table of values may be constructed as shown in
Table 49.15. The working has been shown for the
first two rows.
Let n = 1 to determine y 2 :
2. k 1 = f (x 1 , y 1 ) = f (0.1, 2.205171); since
dy
= y − x, f (0.1, 2.205171)
dx
= 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
)
0.1
,2.205171 +
2 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
)
0.1
,2.205171 +
2 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. y n+1 = y n + h 6 {k 1 + 2k 2 + 2k 3 + k 4 }
and when n = 1:
y 2 = y 1 + h 6 {k 1 + 2k 2 + 2k 3 + k 4 }
= 2.205171+ 0.1
6 {2.105171+2(2.160430)
+ 2(2.163193) + 2.221490}
= 2.205171 + 0.1 {12.973907} =2.421403
6
This completes the third row of Table 49.15. In a
similar manner y 3 , y 4 and y 5 can be calculated and
the results are as shown in Table 49.15. Such a table
is best produced by using a spreadsheet, such as
Microsoft Excel.
This problem is the same as problem 3, page 463
which used Euler’s method, and problem 4, page 465
which used the improved Euler’s method, and a comparison
of results can be made.
The differential equation dy = y − x may be solved
dx
analytically using the integrating factor method of
chapter 48, with the solution:
y = x + 1 + e x
Substituting values of x of 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 below.
Table 49.15
n x n k 1 k 2 k 3 k 4 y n
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