Higher Engineering Mathematics, Sixth Edition

Numerical methods for first order differential equations 475

The percentage error in the Runge-Kutta method
when, say,x= 1 .6is:
(
5. 348811636 − 5. 348817
5. 348811636

)

×100%=− 0 .0001%

From Problem 6, page 469, when x=1.6, the per-
centage error for the Euler method was 0.688%, and
for the Euler-Cauchy method−0.048%. Clearly, the
Runge-Kutta method is the mostaccurate of the three
methods.

Now try the following exercise

Exercise 186 Further problemson the

Runge-Kutta method

1. Apply the Runge-Kutta method to solve the
   differential equation:

dy

dx

= 3 −

y

x

for the range

1.0(0.1)1.5, given that the initial conditions

thatx=1wheny=2.

[see Table 49.19]

Table 49.19

n xn yn

0 1.0 2.0

1 1.1 2.104545

2 1.2 2.216667

3 1.3 2.334615

4 1.4 2.457143

5 1.5 2.533333

2. Obtain a numerical solution of the differential
   equation:

1

x

dy

dx

+ 2 y=1 using the Runge-

Kuttamethod in the rangex=0(0.2)1.0, given

the initial conditions thatx=0wheny=1.

[see Table 49.20]

Table 49.20

n xn yn

0 0 1.0

1 0.2 0.980395

2 0.4 0.926072

3 0.6 0.848838

4 0.8 0.763649

5 1.0 0.683952

3.(a) The differential equation:

dy

dx

+ 1 =−

y

x

has the initial conditions that y=1at

x=2. Produce a numerical solution of the

differential equation, correct to 6 decimal

places, using the Runge-Kutta method in

the rangex=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 49.21 (b) no error]

Table 49.21

n xn yn

0 2.0 1.0

1 2.1 0.854762

2 2.2 0.718182

3 2.3 0.589130

4 2.4 0.466667

5 2.5 0.340000