Higher Engineering Mathematics, Sixth Edition

466 Higher Engineering Mathematics

Table 49.5

x y

0 1

0.2 1

0.4 0.96

0.6 0.8864

0.8 0.793664

1.0 0.699692

(b) If the solution of the differential equa-

tion by an analytical method is given

byy=

4

x

−

x

2

, determine the percentage

error atx= 2. 2

[(a) see Table 49.6 (b) 1.206%]

Table 49.6

x y

2.0 1

2.1 0.85

2.2 0.709524

2.3 0.577273

2.4 0.452174

2.5 0.333334

4. UseEuler’smethodtoobtainanumerical solu-
   tion of the differential equation

dy

dx

=x−

2 y

x

,

given the initial conditions thaty=1when

x=2, in the rangex= 2. 0 ( 0. 2 ) 3 .0.

If the solution of the differential equation is

given byy=

x^2

4

, determine the percentage

error by using Euler’s method whenx= 2. 8

[see Table 49.7, 1.596%]

Table 49.7

x y

2.0 1

2.2 1.2

2.4 1.421818

2.6 1.664849

2.8 1.928718

3.0 2.213187

49.4 An improved Euler method

In Euler’s method of Section 49, the gradient(y′) 0 at

P(x 0 ,y 0 )in Fig. 49.9 across the whole intervalhis used

to obtain an approximate value ofy 1 at pointQ.QRin

Fig. 49.9 is the resulting error in the result.

y

P

x 0 x 1 x

Q

R

y 0

0

h

Figure 49.9

In an improved Euler method, called theEuler-Cauchy

method, the gradient atP(x 0 ,y 0 )across half the interval

is used and then continues with a line whose gradient

approximates to the gradient of the curve atx 1 ,shown

in Fig. 49.10.

LetyP 1 be the predicted value at pointRusing Euler’s

method, i.e. lengthRZ,where

yP 1 =y 0 +h(y′) 0 (3)

The error shown asQTin Fig. 49.10 is now less

than the errorQRused in the basic Euler method and

the calculated results will be of greater accuracy. The

y

P S

x 0 x 1 x

Q

R

Z

T

0

h

x 0112 h

Figure 49.10