17

Eigenfunction methods for

differential equations

In the previous three chapters we dealt with the solution of differential equations

of ordernby two methods. In one method, we foundnindependent solutions

of the equation and then combined them, weighted with coefficients determined

by the boundary conditions; in the other we found solutions in terms of series

whose coefficients were related by (in general) ann-term recurrence relation and

thence fixed by the boundary conditions. For both approaches the linearity of the

equation was an important or essential factor in the utility of the method, and

in this chapter our aim will be to exploit the superposition properties of linear

differential equations even further.

We will be concerned with the solution of equations of the inhomogeneous

form

Ly(x)=f(x), (17.1)

wheref(x) is a prescribed or general functionandthe boundary conditions to

be satisfied by the solutiony=y(x), for example at the limitsx=aandx=b,

are given. The expressionLy(x) stands for a linear differential operatorLacting

upon the functiony(x).

In general, unlessf(x) is both known and simple, it will not be possible to

find particular integrals of (17.1), even if complementary functions can be found

that satisfyLy= 0. The idea is therefore to exploit the linearity ofLby building

up the required solutiony(x)asasuperposition, generally containing an infinite

number of terms, of some set of functions{yi(x)}that each individually satisfy

the boundary conditions. Clearly this brings in a quite considerable complication

but since, within reason, we may select the set of functions to suit ourselves, we

can obtain sizeable compensation for this complication. Indeed, if the set chosen

is one containing functions that, when acted upon byL, produce particularly

simple results then we can ‘show a profit’ on the operation. In particular, if the