Mathematical Methods for Physics and Engineering : A Comprehensive Guide

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS

integrating factor. This is given, as in equation (17.39), by

F(x)=exp

[∫x

c−(a+b+1)u−1+2u

u(1−u)

du

]

=exp

[∫x

c− 1 −(a+b−1)u

u(1−u)

du

]

=exp

[∫x(

c− 1

1 −u

+

c− 1

u

−

a+b− 1

1 −u

du

)]

=exp[(a+b−c)ln(1−x)+(c−1) lnx]

=xc−^1 (1−x)a+b−c.

When the equation is multiplied through byF(x)ittakestheform

[

xc(1−x)a+b−c+1y′

]′

−abxc−^1 (1−x)a+b−cy=0.

Now, for the corresponding Sturm–Liouville operator to be Hermitian, the conditions
to be imposed are as follows.

(i) The boundary condition (17.37); ifc>0anda+b−c+1>0, this is satisfied

automatically for 0≤x≤1, which is thus the natural interval in this case.

(ii) The weight functionxc−^1 (1−x)a+b−cmust be finite and not change sign in the

interval 0≤x≤1. This means that both exponents in it must be positive, i.e.

c− 1 >0anda+b−c>0.

Putting together the conditions on the parameters gives the double inequalitya+b>c>
1.

Finally, we consider Bessel’s equation,

x^2 y′′+xy′+(x^2 −ν^2 )y=0,

which may be converted into Sturm–Liouville form, but only in a somewhat

unorthodox fashion. It is conventional first to divide the Bessel equation byx

and then to change variables to ̄x=x/α. In this case, it becomes

̄xy′′(α ̄x)+y′(α ̄x)−

ν^2

̄x

y(αx ̄)+α^2 ̄xy(α ̄x)=0, (17.41)

where a prime now indicates differentiation with respect to ̄x. Dropping the bars

on the independent variable, we thus have

[xy′(αx)]′−

ν^2

x

y(αx)+α^2 xy(αx)=0, (17.42)

whichisinSLformwithp(x)=x,q(x)=−ν^2 /x,ρ(x)=xandλ=α^2 .It

should be noted, however, that in this case the eigenvalue (actually its square

root) appears in the argument of the dependent variable.