Higher Engineering Mathematics

Differential equations

49

Numerical methods for first order

differential equations

49.1 Introduction

Not all first order differential equations may be

solved by separating the variables (as in Chapter 46)

or by the integrating factor method (as in Chap-

ter 48). A number of other analytical methods of

solving differential equations exist. However the

differential equations that can be solved by such

analytical methods is fairly restricted.

Where a differential equation and known bound-

ary conditions are given, an approximate solution

may be obtained by applying anumerical method.

There are a number of such numerical methods avail-

able and the simplest of these is calledEuler’s

method.

49.2 Euler’s method

From Chapter 8, Maclaurin’s series may be stated as:

f(x)=f(0)+xf′(0)+

x^2

2!

f′′(0)+···

Hence at some pointf(h) in Fig. 51.1:

f(h)=f(0)+hf′(0)+

h^2

2!

f′′(0)+···

y

P

0

h

x

f(h)

Q y = f(x)

f(0)

Figure 49.1

If they-axis and origin are movedaunits to the left,

as shown in Fig. 49.2, the equation of the same curve

relative to the new axis becomesy=f(a+x) and the

function value atPisf(a).

y

P

h

f(a) f(a+x)

Q y = f(a+x)

(^0) x
a
Figure 49.2
At pointQin Fig. 49.2:
f(a+h)=f(a)+hf′(a)+
h^2
2!
f′′(a)+··· (1)
which is a statement calledTaylor’s series.
Ifhis the interval between two new ordinatesy 0
andy 1 , as shown in Fig. 49.3, and iff(a)=y 0 and
y 1 =f(a+h), then Euler’s method states:
f(a+h)=f(a)+hf′(a)
i.e. y 1 =y 0 +h(y′) 0 (2)
0 a (a+h) x
y
P
Q
y 0 y 1
y = f(x)
h
Figure 49.3