Mathematical Methods for Physics and Engineering : A Comprehensive Guide

17.4 STURM–LIOUVILLE EQUATIONS

(ii) From (17.39), the required integrating factor is

F(x)=exp

(∫x

− 1 du

)

=exp(−x).

Thus, the Laguerre equation becomes

xe−xy′′+(1−x)e−xy′+νe−xy=(xe−xy′)′+νe−xy=0,

which is in SL form withp(x)=xe−x,q(x)=0,ρ(x)=e−xandλ=ν.

(iii) From (17.39), the required integrating factor is

F(x)=exp

(∫x

− 2 udu

)

=exp(−x^2 ).

Thus, the Hermite equation becomes

e−x

2

y′′− 2 xe−x

2

y′+2νe−x

2

y=(e−x

2

y′)′+2νe−x

2

y=0,

which is in SL form withp(x)=e−x

2

,q(x)=0,ρ(x)=e−x

2

andλ=2ν.

From thep(x) entries in table 17.1, we may read off the natural interval over

which the corresponding Sturm–Liouville operator (17.35) is Hermitian; in each

case this is given by [a, b], wherep(a)=p(b) = 0. Thus, the natural interval

for the Legendre equation, the associated Legendre equation and the Chebyshev

equation is [− 1 ,1]; for the Laguerre and associated Laguerre equations the

interval is [0,∞]; and for the Hermite equation it is [−∞,∞]. In addition, from

(17.37), one sees that for the simple harmonic equation one requires only that

[a, b]=[x 0 ,x 0 +2π]. We also note that, as required, the weight function in each

case is finite and non-negative over the natural interval. Occasionally, a little more

care is required when determining the conditions for a Sturm–Liouville operator

of the form (17.35) to be Hermitian over some natural interval, as is illustrated

in the following example.

Express the hypergeometric equation,

x(1−x)y′′+[c−(a+b+1)x]y′−aby=0,

in Sturm–Liouville form. Hence determine the natural interval over which the resulting

Sturm–Liouville operator is Hermitian and the corresponding conditions that one must im-

pose on the parametersa,bandc.

As usual for an equation not already in SL form, we first determine the appropriate