EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS
set consists of those functionsyifor which
Lyi(x)=λiyi(x), (17.2)whereλiis a constant (and which satisfy the boundary conditions), then a distinct
advantage may be obtained from the manoeuvre because all the differentiation
will have disappeared from (17.1).
Equation (17.2) is clearly reminiscent of the equation satisfied by theeigenvec-torsxiof a linear operatorA, namely
Axi=λixi, (17.3)whereλiis a constant and is called theeigenvalueassociated withxi. By analogy,
in the context of differential equations a functionyi(x) satisfying (17.2) is called
aneigenfunctionof the operatorL(under the imposed boundary conditions) and
λiis then called the eigenvalue associated with the eigenfunctionyi(x). Clearly,
the eigenfunctionsyi(x)ofLare only determined up to an arbitrary scale factor
by (17.2).
Probably the most familiar equation of the form (17.2) is that which describesa simple harmonic oscillator, i.e.
Ly≡−d^2 y
dt^2=ω^2 y, whereL≡−d^2 /dt^2. (17.4)Imposing the boundary condition that the solution is periodic with periodT,
the eigenfunctions in this case are given byyn(t)=Aneiωnt,whereωn=2πn/T,
n=0,± 1 ,± 2 ,...and theAnare constants. The eigenvalues areω^2 n=n^2 ω^21 =
n^2 (2π/T)^2. (Sometimesωnis referred to as the eigenvalue of this equation, but
we will avoid such confusing terminology here.)
We may discuss a somewhat wider class of differential equations by consideringa slightly more general form of (17.2), namely
Lyi(x)=λiρ(x)yi(x), (17.5)whereρ(x)isaweight function. In many applicationsρ(x) is unity for allx,in
which case (17.2) is recovered; in general, though, it is a function determined by
the choice of coordinate system used in describing a particular physical situation.
The only requirement onρ(x) is that it is real and does not change sign in the
rangea≤x≤b, so that it can, without loss of generality, be taken to be non-
negative throughout; of course,ρ(x) must be the same function for all values of
λi. A functionyi(x) that satisfies (17.5) is called an eigenfunction of the operator
Lwith respect to the weight functionρ(x).
This chapter will not cover methods used to determine the eigenfunctions of(17.2) or (17.5), since we have discussed those in previous chapters, but, rather, will
use the properties of the eigenfunctions to solve inhomogeneous equations of the
form (17.1). We shall see later that the sets of eigenfunctionsyi(x) of a particular