Higher Engineering Mathematics

464 DIFFERENTIAL EQUATIONS

0 0.1 0.2 0.3 0.4 0.5 x

2.5

3.0

y

2.0

Figure 49.8

Euler’s method of numerical solution of differential
equations is simple, but approximate. The method is
most useful when the intervalhis small.

Now try the following exercise.

Exercise 185 Further problems on Euler’s

method

1. Use Euler’s method to obtain a numer-
   ical solution of the differential equation
   dy
   dx

= 3 −

y

x

, with the initial conditions that

x=1 wheny=2, for the rangex= 1 .0to

x= 1 .5 with intervals of 0.1. Draw the graph

of the solution in this range.

[see Table 49.4]

Table 49.4

xy

1.0 2

1.1 2.1

1.2 2.209091

1.3 2.325000

1.4 2.446154

1.5 2.571429

2. Obtain a numerical solution of the differen-

tial equation

1

x

dy

dx

+ 2 y=1, given the initial

conditions thatx=0 wheny=1, in the range

x=0(0.2)1.0. [see Table 49.5]

Table 49.5

xy

01

0.2 1

0.4 0.96

0.6 0.8864

0.8 0.793664

1.0 0.699692

3. (a) The differential equation

dy

dx

+ 1 =−

y

x

has the initial conditions thaty=1at

x=2. Produce a numerical solution of

the differential equation in the range

x= 2 .0(0.1)2.5.

(b) If the solution of the differential equa-

tion by an analytical method is given by

y=

4

x

−

x

2

, determine the percentage error

atx= 2 .2.

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

Table 49.6

xy

2.0 1

2.1 0.85

2.2 0.709524

2.3 0.577273

2.4 0.452174

2.5 0.333334

4. Use Euler’s method to obtain a numer-
   ical solution of the differential equation
   dy
   dx

=x−

2 y

x

, given the initial conditions

that y=1 when x=2, in the range

x= 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%]