Higher Engineering Mathematics, Sixth Edition

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

1. 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

2. 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

4. 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