Higherorder equations

NUMERICAL METHODS
(ii) To order h4 ,
yi+1 = yi + 16 (c1 + 2c2 + 2c3 + c4 ),
(27.80)
where
c1 = hf(xi , yi ),
c2 = hf(xi + 12 h, yi + 12 c1 ),
c3 = hf(xi + 12 h, yi + 12 c2 ),
c4 = hf(xi + h, yi + c3 ).
27.6.5 Isoclines
The final method to be described for first-order differential equations is not so
much numerical as graphical, but since it is sometimes useful it is included here.
The method, known as that of isoclines, involves sketching for a number of
values of a parameter c those curves (the isoclines) in the xy-plane along which
f(x, y) = c, i.e. those curves along which dy/dx is a constant of known value. It
should be noted that isoclines are not generally straight lines. Since a straight
line of slope dy/dx at and through any particular point is a tangent to the curve
y = y(x) at that point, small elements of straight lines, with slopes appropriate
to the isoclines they cut, effectively form the curve y = y(x).
Figure 27.6 illustrates in outline the method as applied to the solution of
dy
= −2xy.
dx
(27.81)
The thinner curves (rectangular hyperbolae) are a selection of the isoclines along
which −2xy is constant and equal to the corresponding value of c. The small
cross lines on each curve show the slopes (= c) that solutions of (27.81) must
have if they cross the curve. The thick line is the solution for which y = 1 at
x = 0; it takes the slope dictated by the value of c on each isocline it crosses. The
analytic solution with these properties is y(x) = exp(−x2 ).
27.7 Higher-order equations
So far the discussion of numerical solutions of differential equations has been
in terms of one dependent and one independent variable related by a first-order
equation. It is straightforward to carry out an extension to the case of several
dependent variables y[r] governed by R first-order equations:
dy[r]
= f[r] (x, y[1] , y[2] , . . . , y[R] ),
dx
r = 1, 2, . . . , R.
We have enclosed the label r in brackets so that there is no confusion between,
say, the second dependent variable y[2] and the value y2 of a variable y at the
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27.7 HIGHER-ORDER EQUATIONS
y
1.0
0.8
0.6
y
0.4
0.2
0.2
0.4
0.6
0.8
c
−1.0
−0.8
−0.6
−0.4
−0.2
−0.1
1.0 x
Figure 27.6 The isocline method. The cross lines on each isocline show the
slopes that solutions of dy/dx = −2xy must have at the points where they
cross the isoclines. The heavy line is the solution with y(0) = 1, namely
exp(−x2 ).
second calculational point x2 . The integration of these equations by the methods
discussed in the previous section presents no particular difficulty, provided that
all the equations are advanced through each particular step before any of them
is taken through the following step.
Higher-order equations in one dependent and one independent variable can be
reduced to a set of simultaneous equations, provided that they can be written in
the form
dR y
= f(x, y, y , . . . , y (R−1) ),
dxR
(27.82)
where R is the order of the equation. To do this, a new set of variables p[r] is
defined by
p[r] =
dr y
,
dxr
r = 1, 2, . . . , R − 1.
(27.83)
Equation (27.82) is then equivalent to the following set of simultaneous first-order
equations:
dy
= p[1] ,
dx
dp[r]
= p[r+1] ,
dx
r = 1, 2, . . . , R − 2,
dp[R−1]
= f(x, y, p[1] , . . . , p[R−1] ).
dx
1029
(27.84)