Chapter 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 Chapter 48). A number
of other analytical methods of solving differential equa-
tions exist. However the differential equations that can
be solved by such analytical methods is fairly restricted.
Where a differential equation and known boundary
conditions are given, an approximate solution may be
obtained by applying anumerical method.Thereare
a number of such numerical methods available 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. 49.1:

f(h)=f( 0 )+hf′( 0 )+

h^2

2!

f′′( 0 )+···

If they-axis and origin are movedaunits to the left,
as shown in Fig. 49.2, the equation of the same curve

y

y 5 f(x)

P

h

x

Q

f(0)

f(h)

0

Figure 49.1

y y 5 f(a 1 x)

P

a h

x

Q

f(a) f(a 1 x)

0

Figure 49.2

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

function value atPisf(a).

At pointQin Fig. 49.2:

f(a+h)=f(a)+hf′(a)+

h^2

2!

f′′(a)+··· (1)