Mathematical Methods for Physics and Engineering : A Comprehensive Guide

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS

y=xλ. This leads to an algebraic equation whose solution gives the allowed values

ofλ; the general solution is then the linear superposition of these functions.

15.2.2 Exact equations

Sometimes an ODE may be merely the derivative of another ODE of one order

lower. If this is the case then the ODE is called exact. Thenth-order linear ODE

an(x)

dny

dxn

+···+a 1 (x)

dy

dx

+a 0 (x)y=f(x), (15.41)

is exact if the LHS can be written as a simple derivative, i.e. if

an(x)

dny

dxn

+···+a 0 (x)y=

d

dx

[

bn− 1 (x)

dn−^1 y

dxn−^1

+···+b 0 (x)y

]

. (15.42)

It may be shown that, for (15.42) to hold, we require

a 0 (x)−a′ 1 (x)+a′′ 2 (x)−···+(−1)na(nn)(x)=0, (15.43)

where the prime again denotes differentiation with respect tox. If (15.43) is

satisfied then straightforward integration leads to a new equation of one order

lower. If this simpler equation can be solved then a solution to the original

equation is obtained. Of course, if the above process leads to an equation that is

itself exact then the analysis can be repeated to reduce the order still further.

Solve

(1−x^2 )

d^2 y

dx^2

− 3 x

dy

dx

−y=1. (15.44)

Comparing with (15.41), we havea 2 =1−x^2 ,a 1 =− 3 xanda 0 =−1. It is easily shown
thata 0 −a′ 1 +a′′ 2 = 0, so (15.44) is exact and can therefore be written in the form

d

dx

[

b 1 (x)

dy

dx

+b 0 (x)y

]

=1. (15.45)

Expanding the LHS of (15.45) we find

d

dx

(

b 1

dy

dx

+b 0 y

)

=b 1

d^2 y

dx^2

+(b′ 1 +b 0 )

dy

dx

+b′ 0 y. (15.46)

Comparing (15.44) and (15.46) we find

b 1 =1−x^2 ,b′ 1 +b 0 =− 3 x, b′ 0 =− 1.

These relations integrate consistently to giveb 1 =1−x^2 andb 0 =−x, so (15.44) can be
written as
d
dx

[

(1−x^2 )

dy

dx

−xy

]

=1. (15.47)

Integrating (15.47) gives us directly the first-order linear ODE

dy

dx

−

( x

1 −x^2

)

y=

x+c 1

1 −x^2

,

which can be solved by the method of subsection 14.2.4 and has the solution

y=

c 1 sin−^1 x+c 2

√

1 −x^2

− 1 .