Higher Engineering Mathematics

468 DIFFERENTIAL EQUATIONS

For line 6,x 1 = 2. 0

yP 1 =y 0 +h(y′) 0

= 5. 85212176 + 0 .2(2.54787824)

= 6. 361697408

yC 1 =y 0 +^12 h[(y′) 0 +(3+ 3 x 1 −yP 1 )]

= 5. 85212176 +^12 (0.2)[2. 54787824

+(3+3(2.0)− 6 .361697408)]

=6.370739843

Problem 6. Using the integrating factor

method the solution of the differential equa-

tion

dy

dx

=3(1+x)−y of Problem 5 is

y= 3 x+e^1 −x. When x= 1 .6, compare the

accuracy, correct to 3 decimal places, of the

Euler and the Euler-Cauchy methods.

Whenx= 1 .6,y= 3 x+e^1 −x=3(1.6)+e^1 −^1.^6 =
4. 8 +e−^0.^6 = 5 .348811636.
From Table 49.1, page 461, by Euler’s method,
whenx= 1 .6,y= 5. 312
% error in the Euler method

=

(

5. 348811636 − 5. 312

5. 348811636

)

×100%

=0.688%

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 accu-
rate than the Euler method whenx= 1 .6.

Now try the following exercise.

Exercise 186 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=1 wheny= 2.

[see Table 49.12]

Table 49.12

xy 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, when
x= 1 .3 using (a) Euler’s method (see
Table 49.4, page 464), 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 .5 in incre-

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 byy=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

xy y′

01 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