Mathematical Methods for Physics and Engineering : A Comprehensive Guide

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS

15.2.5 Green’s functions

The Green’s function method of solving linear ODEs bears a striking resemblance

to the method of variation of parameters discussed in the previous subsection;

it too requires knowledge of the entire complementary function in order to find

the particular integral and therefore the general solution. The Green’s function

approach differs, however, since once the Green’s function for a particular LHS

of (15.1) and particular boundary conditions has been found, then the solution

foranyRHS (i.e. anyf(x)) can be written down immediately, albeit in the form

of an integral.

Although the Green’s function method can be approached by considering the

superposition of eigenfunctions of the equation (see chapter 17) and is also

applicable to the solution of partial differential equations (see chapter 21), this

section adopts a more utilitarian approach based on the properties of the Dirac

delta function (see subsection 13.1.3) and deals only with the use of Green’s

functions in solving ODEs.

Let us again consider the equation

an(x)

dny

dxn

+···+a 1 (x)

dy

dx

+a 0 (x)y=f(x), (15.58)

but for the sake of brevity we now denote the LHS byLy(x), i.e. as a linear

differential operator acting ony(x). Thus (15.58) now reads

Ly(x)=f(x). (15.59)

Let us suppose that a functionG(x, z) (theGreen’s function) exists such that the

general solution to (15.59), which obeys some set of imposed boundary conditions

in the rangea≤x≤b, is given by

y(x)=

∫b

a

G(x, z)f(z)dz, (15.60)

wherezis the integration variable. If we apply the linear differential operatorL

to both sides of (15.60) and use (15.59) then we obtain

Ly(x)=

∫b

a

[LG(x, z)]f(z)dz=f(x). (15.61)

Comparison of (15.61) with a standard property of the Dirac delta function (see

subsection 13.1.3), namely

f(x)=

∫b

a

δ(x−z)f(z)dz,

fora≤x≤b, shows that for (15.61) to hold for any arbitrary functionf(x), we

require (fora≤x≤b)that

LG(x, z)=δ(x−z), (15.62)