NUMERICAL METHODS FOR FIRST ORDER DIFFERENTIAL EQUATIONS 473
I
Table 49.18
Euler’s Euler-Cauchy Runge-Kutta
method method method Exact value
x y y y y= 3 x+e^1 −x
1.0 4 4 4 4
1.2 4.4 4.42 4.418733 4.418730753
1.4 4.84 4.8724 4.870324 4.870320046
1.6 5.312 5.351368 5.348817 5.348811636
1.8 5.8096 5.85212176 5.849335 5.849328964
2.0 6.32768 6.370739847 6.367886 6.367879441
Table 49.19
n xn yn
0 1.0 2.0
1 1.1 2.104545
2 1.2 2.216667
3 1.3 2.334615
4 1.4 2.457143
5 1.5 2.533333
- Obtain a numerical solution of the differential
equation:
1
x
dy
dx
+ 2 y=1 using the Runge-
Kutta method in the range x=0(0.2)1.0,
given the initial conditions thatx=0 when
y=1. [see Table 49.20]
Table 49.20
n xn yn
0 0 1.0
1 0.2 0.980395
2 0.4 0.926072
3 0.6 0.848838
4 0.8 0.763649
5 1.0 0.683952
- (a) The differential equation:
dy
dx
+ 1 =−
y
x
has the initial conditions that y=1at
x=2. Produce a numerical solution of the
differential equation, correct to 6 decimal
places, using the Runge-Kutta method in
the rangex=2.0(0.1)2.5.
(b) If the solution of the differential equa-
tion by an analytical method is given by:
y=
4
x
−
x
2
determine the percentage error
atx=2.2.
[(a) see Table 49.21 (b) no error]
Table 49.21
n xn yn
0 2.0 1.0
1 2.1 0.854762
2 2.2 0.718182
3 2.3 0.589130
4 2.4 0.466667
5 2.5 0.340000
