Mathematical Methods for Physics and Engineering : A Comprehensive Guide

15.3 GENERAL ORDINARY DIFFERENTIAL EQUATIONS

but also write

d^2 y

dx^2

=

dp

dx

=

dy

dx

dp

dy

=p

dp

dy

d^3 y

dx^3

=

d

dx

(

p

dp

dy

)

=

dy

dx

d

dy

(

p

dp

dy

)

=p^2

d^2 p

dy^2

+p

(

dp

dy

) 2

, (15.78)

and so on for higher-order derivatives. This leads to an equation of one order

lower.

Solve

1+y

d^2 y

dx^2

+

(

dy

dx

) 2

=0. (15.79)

Making the substitutionsdy/dx=pandd^2 y/dx^2 =p(dp/dy) we obtain the first-order
ODE

1+yp

dp

dy

+p^2 =0,

which is separable and may be solved as in subsection 14.2.1 to obtain

(1 +p^2 )y^2 =c 1.

Usingp=dy/dxwe therefore have

p=

dy

dx

=±

√

c^21 −y^2

y^2

,

which may be integrated to give the general solution of (15.79); after squaring this reads

(x+c 2 )^2 +y^2 =c^21 .

Solution method.If the ODE does not containxexplicitly then substitutep=

dy/dx, along with the relations for higher derivatives given in (15.78), to obtain an

equation of one order lower, which may prove easier to solve.

15.3.3 Non-linear exact equations

As discussed in subsection 15.2.2, an exact ODE is one that can be obtained by

straightforward differentiation of an equation of one order lower. Moreover, the

notion of exact equations is useful for both linear and non-linear equations, since

an exact equation can be immediately integrated. It is possible, of course, that

the resulting equation may itself be exact, so that the process can be repeated.

In the non-linear case, however, there is no simple relation (such as (15.43) for

the linear case) by which an equation can be shown to be exact. Nevertheless, a

general procedure does exist and is illustrated in the following example.