470 Higher Engineering Mathematics
From Table 49.11 of Problem 5, by the Euler-Cauchy
method, whenx= 1 .6,y= 5. 351368
% error in the Euler-Cauchy method
=
(
5. 348811636 − 5. 351368
5. 348811636
)
×100%
=−0.048%
The Euler-Cauchy method is seen to be more accurate
than the Euler method whenx= 1 .6.
Now try the following exercise
Exercise 185 Further problems on an
improved Euler method
- Apply the Euler-Cauchy method to solve the
differential equation
dy
dx
= 3 −
y
x
for the range 1.0(0.1)1.5, given the initial
conditions thatx=1wheny= 2.
[see Table 49.12]
Table 49.12
x y y′
1.0 2 1
1.1 2.10454546 1.08677686
1.2 2.216666672 1.152777773
1.3 2.33461539 1.204142008
1.4 2.457142859 1.2448987958
1.5 2.583333335
- Solving the differential equation in Prob-
lem 1 by the integrating factor method gives
y=
3
2
x+
1
2 x
. Determine the percentage error,
correct to 3 significant figures, whenx= 1. 3
using (a) Euler’s method (see Table 49.4,
page 465), and (b) the Euler-Cauchy method.
[(a) 0.412% (b) 0.000000214%]
3.(a) Apply the Euler-Cauchy method to solve
the differential equation
dy
dx
−x=y
for the rangex=0tox= 0 .5inincre-
ments of 0.1, given the initial conditions
that whenx=0,y= 1
(b) The solution of the differential equation
in part (a) is given by y=2ex−x−1.
Determine the percentage error, correct to
3 decimal places, whenx= 0. 4
[(a) see Table 49.13 (b) 0.117%]
Table 49.13
x y y′
0 1 1
0.1 1.11 1.21
0.2 1.24205 1.44205
0.3 1.398465 1.698465
0.4 1.581804 1.981804
0.5 1.794893
- Obtain a numerical solution of the differential
equation
1
x
dy
dx
+ 2 y= 1
using the Euler-Cauchy method in the range
x= 0 ( 0. 2 ) 1 .0, given the initial conditions that
x=0wheny= 1.
[see Table 49.14]
Table 49.14
x y y′
0 1 0
0.2 0.99 −0.196
0.4 0.958336 −0.3666688
0.6 0.875468851 −0.450562623
0.8 0.784755575 −0.45560892
1.0 0.700467925