Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.7 HIGHER-ORDER EQUATIONS

0. 2

0. 2

0. 4

0. 4

0. 6

0. 6

0. 8

0. 8

1. 0

1. 0

c

y

y

x

− 1. 0

− 0. 8

− 0. 6

− 0. 4

− 0. 2

− 0. 1

Figure 27.6 The isocline method. The cross lines on each isocline show the

slopes that solutions ofdy/dx=− 2 xymust have at the points where they

cross the isoclines. The heavy line is the solution withy(0) = 1, namely

exp(−x^2 ).

second calculational pointx 2. 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

dRy

dxR

=f(x, y, y′,...,y(R−1)), (27.82)

whereRis the order of the equation. To do this, a new set of variablesp[r]is

defined by

p[r]=

dry

dxr

,r=1, 2 ,...,R− 1. (27.83)

Equation (27.82) is then equivalent to the following set of simultaneous first-order

equations:

dy

dx

=p[1],

dp[r]

dx

=p[r+1],r=1, 2 ,...,R− 2 , (27.84)

dp[R−1]

dx

=f(x, y, p[1],...,p[R−1]).