Mathematical Methods for Physics and Engineering : A Comprehensive Guide

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS

i.e. the Green’s functionG(x, z)must satisfy the original ODE with the RHS set

equal to a delta function.G(x, z) may be thought of physically as the response of

a system to a unit impulse atx=z.

In addition to (15.62), we must impose two further sets of restrictions on

G(x, z). The first is the requirement that the general solutiony(x) in (15.60) obeys

the boundary conditions. Forhomogeneousboundary conditions, in whichy(x)

and/or its derivatives are required to bezeroat specified points, this is most

simply arranged by demanding thatG(x, z) itself obeys the boundary conditions

when it is considered as a function ofxalone; if, for example, we require

y(a)=y(b) = 0 then we should also demandG(a, z)=G(b, z) = 0. Problems

having inhomogeneous boundary conditions are discussed at the end of this

subsection.

The second set of restrictions concerns the continuity or discontinuity ofG(x, z)

and its derivatives atx=zand can be found by integrating (15.62) with respect

toxover the small interval [z−, z+] and taking the limit as→0. We then

obtain

lim

→ 0

∑n

m=0

∫z+

z−

am(x)

dmG(x, z)

dxm

dx= lim

→ 0

∫z+

z−

δ(x−z)dx=1. (15.63)

SincednG/dxnexists atx=zbut with value infinity, the (n−1)th-order derivative

must have a finite discontinuity there, whereas all the lower-order derivatives,

dmG/dxmform<n−1, must be continuous at this point. Therefore the terms

containing these derivatives cannot contribute to the value of the integral on

the LHS of (15.63). Noting that, apart from an arbitrary additive constant,∫

(dmG/dxm)dx=dm−^1 G/dxm−^1 , and integrating the terms on the LHS of (15.63)

by parts we find

lim

→ 0

∫z+

z−

am(x)

dmG(x, z)

dxm

dx= 0 (15.64)

form=0ton−1. Thus, since only the term containingdnG/dxncontributes to

the integral in (15.63), we conclude, after performing an integration by parts, that

lim

→ 0

[

an(x)

dn−^1 G(x, z)

dxn−^1

]z+

z−

=1. (15.65)

Thus we have the furthernconstraints thatG(x, z) and its derivatives up to order

n−2 are continuous atx=zbut thatdn−^1 G/dxn−^1 has a discontinuity of 1/an(z)

atx=z.

Thus the properties of the Green’s functionG(x, z)forannth-order linear ODE

may be summarised by the following.

(i)G(x, z) obeys the original ODE but withf(x) on the RHS set equal to a

delta functionδ(x−z).