Higher Engineering Mathematics

466 DIFFERENTIAL EQUATIONS

Table 49.8

xy y′

1. 0 2 2


2. 0.1 2.205 2.105


3. 0.2 2.421025 2.221025


4. 0.3 2.649232625 2.349232625


5. 0.4 2.89090205 2.49090205


6. 0.5 3.147446765

For line 3,x 1 = 0. 2

yP 1 =y 0 +h(y′) 0 = 2. 205 +(0.1)(2.105)

= 2. 4155

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

= 2. 205 +^12 (0.1)[2. 105 +(2. 4155 − 0 .2)]

=2.421025

(y′) 0 =yC 1 −x 1 = 2. 421025 − 0. 2 = 2. 221025

For line 4,x 1 = 0. 3

yP 1 =y 0 +h(y′) 0

= 2. 421025 +(0.1)(2.221025)

= 2. 6431275

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

= 2. 421025 +^12 (0.1)[2. 221025

+(2. 6431275 − 0 .3)]

=2.649232625

(y′) 0 =yC 1 −x 1 = 2. 649232625 − 0. 3

= 2. 349232625

For line 5,x 1 = 0. 4

yP 1 =y 0 +h(y′) 0

= 2. 649232625 +(0.1)(2.349232625)

= 2. 884155887

Table 49.9

Euler method Euler-Cauchy method Exact value

xy y y=x+ 1 +ex

1. 0 2 2 2


2. 0.1 2.2 2.205 2.205170918


3. 0.2 2.41 2.421025 2.421402758


4. 0.3 2.631 2.649232625 2.649858808


5. 0.4 2.8641 2.89090205 2.891824698


6. 0.5 3.11051 3.147446765 3.148721271

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

= 2. 649232625 +^12 (0.1)[2. 349232625

+(2. 884155887 − 0 .4)]

=2.89090205

(y′) 0 =yC 1 −x 1 = 2. 89090205 − 0. 4

= 2. 49090205

For line 6,x 1 = 0. 5

yP 1 =y 0 +h(y′) 0

= 2. 89090205 +(0.1)(2.49090205)

= 3. 139992255

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

= 2. 89090205 +^12 (0.1)[2. 49090205

+(3. 139992255 − 0 .5)]

=3.147446765

Problem 4 is the same example as Problem 3 and

Table 49.9 shows a comparison of the results, i.e.

it compares the results of Tables 49.3 and 49.8.

dy

dx

=y−xmay be solved analytically by the inte-

grating factor method of Chapter 48 with the solution

y=x+ 1 +ex. Substituting values ofxof 0, 0.1,

0 .2,...give the exact values shown in Table 49.9.

The percentage error for each method for each

value ofxis shown in Table 49.10. For example

whenx= 0 .3,

% error with Euler method

=

(

actual−estimated

actual

)

×100%

=

(

2. 649858808 − 2. 631

2. 649858808

)

×100%

=0.712%