20.9 First-order differential equations 587
A numerical method for solving a differential equation is equivalent to starting at
a particular point in the direction field and following the field lines. A number of
methods are available for the solution of most types of differential equation met in the
physical sciences, and most of these have their origin in the Taylor expansion (7.24).
Given the valuey(x
0) 1 = 1 y
0of the functiony(x)atx 1 = 1 x
0, the value at the neighbouring
pointx
11 = 1 x
01 + 1 his
(20.50)
Then, becausey′(x
0) 1 = 1 f(x
0, y
0)by (20.49),
or
y
11 = 1 y
01 + 1 hF
0(20.51)
whereF
01 = 1 (y
11 − 1 y
0) 2 (x
11 − 1 x
0)is the slope of the line joining the points 0 and 1 on the
graph ofy(x)(the average slope of the curve). The different numerical methods use
different way of estimating this slope, leading to the recursion relation
y
n+ 11 = 1 y
n1 + 1 hF
n(20.52)
wherex
n1 = 1 x
01 + 1 nhandy
n1 ≈ 1 y(x
n). We consider first the simplest possible method,
Euler’s method.
Euler’s method
Euler’s method is to truncate the Taylor expansion after the second term,
y(x
1) 1 ≈ 1 y(x
0) 1 + 1 hy′(x
0)
so that F
0has been approximated by the slope at the initial point (x
0, y
0). An
approximate value ofy(x
1)is then
y
11 = 1 y
01 + 1 hf(x
0, y
0) (20.53)
and this provides us with a simple step-by-step procedure whereby an approximate
solution of the equation is obtained by repeated application of the recursion
y
n+ 11 = 1 y
n1 + 1 hf(x
n, y
n), n 1 = 1 0, 1, 2, = (20.54)
wherex
n1 = 1 x
01 + 1 nhand y
n1 ≈ 1 y(x
n). At each step the slope of the line from point n
to point (n 1 + 11 ) is that of the direction field line at point n, and the procedure is
illustrated in Figure 20.11.
yx yx hf x y() () ( ) hy x()
10 00201
2
=+ ,+′′ +
yx yx hy x() () () hy x()
10 0201
2
=+′+′′ +