# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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x/ + y + Xxy = 0, (6)

or

(x/y = Xxy, (7)

on the interval 0 < x < 1, and suppose that X > 0. This equation arises in the study of free vibrations of a circular elastic membrane, and is discussed further in Section 11.5. If we introduce the new independent variable t defined by t = */X x, then

dy r- dy d2 y d2 y

= 41^, = X 2

dx dt dx2 dt2

Hence Eq. (6) becomes

t d2 y r- dy t n X~~2 + *dX + Xy = 0, VX dt2 dt Jx

or, if we cancel the common factor VX in each term

d2 y dy

dt2 dt

d2 y dy

t^T + -4 + ty = 0. (8)

6The substitution t = s/Xx reduces Eq. (4) to the standard form t2y" + ty + (t2 v2)y = 0.

658

Chapter 11. Boundary Value Problems and Sturm Liouville Theory

Equation (8) is Bessels equation of order zero (see Section 5.8). The general solution of Eq. (8) for t > 0 is

/ = c1 J0(t) + c2Y0(t); hence the general solution of Eq. (7) for x > 0 is

J = c1 J0(VXx) + c2 Y0(VXx), (9)

where J0 and 10 denote the Bessel functions of the first and second kinds of order zero. From Eqs. (7) and (13) of Section 5.8 we have

? ( 1 )m -m x2m

J^x) = 1+ J2 2m , x > 0, uo)

m=i 22m (m!)2

r- 2

Y0 (VX x) = -n

2 VXx\ T n ? (-1)m+1 \mxtm

Y + ln - I J0(VXx) + ^-----------2------2---

Y 2 0( ) Z. 22m (m!)2

x > 0, (11)

where Hm = 1 + (1/2) + + (1/m),and y = lim (Hm - ln m). The graphs of y =

J0(x) and y = Y0(x) are given in Figure 5.8.2.

Suppose that we seek a solution of Eq. (7) that also satisfies the boundary conditions

y(0) = 0, (12)

y(1) = 0, (13)

which are typical of those we have met in other problems in this chapter. Since J0 (0) = 1 and Y0(x) ^ -ccas x ^ 0, the condition y(0) = 0 can be satisfied only by choosing

q = c2 = 0 in Eq. (9). Thus the boundary value problem (7), (12), (13) has only the

trivial solution.

One interpretation of this result is that the boundary condition (12) at x = 0 is too restrictive for the differential equation (7). This illustrates the general situation, namely, that at a singular boundary point it is necessary to consider a modified type of boundary condition. In the present problem, suppose that we require only that the solution (9) and its derivative remain bounded. In other words, we take as the boundary condition at x = 0 the requirement

y, y bounded as x ^ 0. (14)

This condition can be satisfied by choosing c2 = 0 in Eq. (9), so as to eliminate the

unbounded solution 10. The second boundary condition, y(1) = 0, then yields

J0(VK) = 0. (15)

It is possible to show7 that Eq. (15) has an infinite set of discrete positive roots,

which yield the eigenvalues 0 < K1 < K2 < ??? < Kn < ??? of the given problem. The corresponding eigenfunctions are

&n(x) = , (16)

7The function J0 is well tabulated; the roots of Eq. (15) can be found in various tables, for example, those in Jahnke and Emde or Abramowitz and Stegun. The first three roots of Eq. (15) are s/k = 2.405, 5.520, and 8.654, respectively, to four significant figures; yKj = (n - 1/4)n for large n.

11.4 Singular Sturm-Liouville Problems

659

determined only up to a multiplicative constant. The boundary value problem (7), (13), and (14) is an example of a singular Sturm-Liouville problem. This example illustrates that if the boundary conditions are relaxed in an appropriate way, then a singular Sturm-Liouville problem may have an infinite sequence of eigenvalues and eigenfunctions, just as a regular Sturm-Liouville problem does.

Because of their importance in applications, it is worthwhile to investigate singular boundary value problems a little further. There are two main questions that are of concern.

1. Precisely what type of boundary conditions can be allowed in a singular Sturm-Liouville problem?

2. To what extent do the eigenvalues and eigenfunctions of a singular problem share the properties of eigenvalues and eigenfunctions of regular Sturm-Liouville problems? In particular, are the eigenvalues real, are the eigenfunctions orthogonal, and can a given function be expanded as a series of eigenfunctions?

Both these questions can be answered by a study of the identity

-1

{L[u]v uL[v]} dx = 0, (17)

J

Jo

which played an essential part in the development of the theory of regular Sturm-Liouville problems. We therefore investigate the conditions under which this relation holds for singular problems, where the integral in Eq. (17) may now have to be examined as an improper integral. To be definite we consider the differential equation (1) and assume that x = 0 is a singular boundary point, but that x = 1 is not. The boundary condition (3) is imposed at the nonsingular boundary point x = 1, but we leave unspecified, for the moment, the boundary condition at x = 0. Indeed, our principal object is to determine what kinds of boundary conditions are allowable at a singular boundary point if Eq. (17) is to hold.

Since the boundary value problem under investigation is singular at x = 0, we choose

> 0 and consider the integral J L [u]v dx, instead of L[u]v dx, as in Section

11.2. Afterwards we let approach zero. Assuming that u and v have at least two continuous derivatives on < x < 1, and integrating twice by parts, we find that

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