# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

**Download**(direct link)

**:**

**359**> 360 361 362 363 364 365 .. 609 >> Next

A systematic discussion of singular Sturm-Liouville problems is quite sophisticated8 indeed, requiring a substantial extension of the methods presented in this book. We restrict ourselves to some examples related to physical applications; in each of these examples it is known that there is an infinite set of discrete eigenvalues.

If a singular Sturm-Liouville problem does have only a discrete set of eigenvalues and eigenfunctions, then Eq. (17) can be used, just as in Section 11.2, to prove that the eigenvalues of such a problem are real, and that the eigenfunctions are orthogonal with respect to the weight function r. The expansion of a given function in terms of a series of eigenfunctions then follows as in Section 11.2.

Such expansions are useful, as in the regular case, for solving nonhomogeneous boundary value problems. The procedure is very similar to that described in Section

11.3. Some examples for ordinary differential equations are indicated in Problems 1 to

4, and some problems for partial differential equations appear in Section 11.5.

For instance, the eigenfunctions ôï (x) = J0 (^fX~n x) of the singular Sturm-Liouville problem

— (xy/)’ = Xxy, 0 < x < 1,

y, / bounded as x ^ 0, y(1) = 0

8See, for example, Chapter 5 of the book by Yosida.

11.4 Singular Sturm Liouville Problems

661

PROBLEMS

satisfy the orthogonality relation

xÔò(x)Ôï(x) dx = 0, m = n (23)

m

0

with respect to the weight function r (x) = x. Then, if f is a given function, we assume that

n= 1

f(x) = E CnJ0(Jknx)- (24)

Multiplying Eq. (24) by xJ0(^xmx) and integrating term by term from x = 0to x = 1 yield

J0 xf(x) J0 (ó[Ê x) dx = ? Cnjo xJ0 (ó[Ê x) J0 (fin x) dx¦ (25)

Because of the orthogonality condition (23), the right side of Eq. (25) collapses to a single term; hence

j xf(x)J0(fi-mx) dx

Jo xJ02(^fKx) dx

(26)

which determines the coefficients in the series (24).

The convergence of the series (24) is established by an extension of Theorem 11.2.4 to cover this case. This theorem can also be shown to hold for other sets of Bessel functions, which are solutions of appropriate boundary value problems, for Legendre polynomials, and for solutions of a number of other singular Sturm-Liouville problems of considerable interest.

It must be emphasized that the singular problems mentioned here are not necessarily typical. In general, singular boundary value problems are characterized by continuous spectra, rather than by discrete sets of eigenvalues. The corresponding sets of eigenfunctions are therefore not denumerable, and series expansions of the type described in Theorem 11.2.4 do not exist. They are replaced by appropriate integral representations.

cm =

1. Find a formal solution of the nonhomogeneous boundary value problem

- (xy) = Ä-xy + f (x),

y, Ó bounded as x ^ 0, y(1) = 0,

where f is a given continuous function on 0 < x < 1, and ä is not an eigenvalue of the corresponding homogeneous problem.

Hint: Use a series expansion similar to those in Section 11.3.

2. Consider the boundary value problem

- (xy) = Xxy,

y, Ó bounded as x ^ 0, Ó (1) = 0.

(a) Show that k0 = 0 is an eigenvalue of this problem corresponding to the eigenfunction ô0(x) = 1. If ê > 0, show formally that the eigenfunctions are given by ôï(x) =

662

Chapter 11. Boundary Value Problems and Sturm-Liouville Theory

J0 (VK,x), where is the nth positive root (in increasing order) of the equation J0 (\/Y) = 0. It is possible to show that there is an infinite sequence of such roots.

(b) Show that if m, n = 0, 1, 2,..., then

(c) Find a formal solution to the nonhomogeneous problem

- (xf)' = ^xy + f(x),

y, y bounded as x ^ 0, Ó (1) = 0,

where f is a given continuous function on 0 < x < 1, and ä is not an eigenvalue of the corresponding homogeneous problem.

3. Consider the problem

where k is a positive integer.

(a) Using the substitution t = -JY x, show that the given differential equation reduces to Bessel’s equation of order k (see Problem 9 of Section 5.8). One solution is Jk(t);a second linearly independent solution, denoted by Yk(t), is unbounded as t ^ 0.

(b) Show formally that the eigenvalues k1,k2,... of the given problem are the squares of the positive zeros of Jk (vY), and that the corresponding eigenfunctions are ôï (x) = Jk(yfY~x). It is possible to show that there is an infinite sequence of such zeros.

(c) Show that the eigenfunctions ôï (x) satisfy the orthogonality relation

**359**> 360 361 362 363 364 365 .. 609 >> Next