1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

5.8 Some Applications of Bessel Functions 333

(

x^3 X′

)′

+λ^2 x^3 X= 0 , a<x<b, (31)

X(a)= 0 , X(b)= 0 , (32)

Y′′−λ^2 Y= 0 , −c<y<c. (33)

Equation (31) may be put in the form

X′′+^3 xX′+λ^2 X= 0 , a<x<b.

By comparing to Eq. (1) we find thatα=−1,γ=1, andp=1 and that the
generalsolutionofEq.(31)is

X(x)=^1 x

(

AJ 1 (λx)+BY 1 (λx)

)

.

Because the pointx=0 is not included in the intervala<x<b,thereisno
problem with boundedness. Instead we must satisfy the boundary conditions
Eq. (32), which after some algebra have the form

AJ 1 (λa)+BY 1 (λa)= 0 ,
AJ 1 (λb)+BY 1 (λb)= 0.
Not bothAandBmay be zero, so the determinant of these simultaneous
equations must be zero:

J 1 (λa)Y 1 (λb)−J 1 (λb)Y 1 (λa)= 0.

Some solutions of the equation are tabulated for various values ofb/a.For
instance, ifb/a= 2 .5, the first three eigenvaluesλ^2 are
( 2. 156
a

) 2

,

( 4. 223

a

) 2

,

( 6. 307

a

) 2

.

We n o w c a n t a k eXnto be

Xn(x)=^1

x

(

Y 1 (λna)J 1 (λnx)−J 1 (λna)Y 1 (λnx)

)

, (34)

and the solution of Eqs. (28)–(30) has the form

u(x,y)=

∑∞

n= 1

anXn(x)

cosh(λny)

cosh(λnc). (35)

Thea’s are chosen to satisfy the boundary conditions Eq. (30), using the or-
thogonality principle∫
b
a

Xn(x)Xm(x)x^3 dx= 0 , n=m.

Notice that Eqs. (31) and (32) make up a regular Sturm–Liouville problem.