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

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is zero.

We use the term singular Sturm-Liouville problem to refer to a certain class of boundary value problems for the differential equation (1) in which the functions p, q, and r satisfy the conditions stated earlier on the open interval 0 < x < 1, but at least

one of these functions fails to satisfy them at one or both of the boundary points. We

also prescribe suitable separated boundary conditions, of a kind to be described in more detail later in this section. Singular problems also occur if the interval is unbounded, for example, 0 < x < to. We do not consider this latter kind of singular problem in this book.

As an example of a singular problem on a finite interval, consider the equation

xy + y + Xxy = 0, (6)

or

— (xy) = 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 nr dy d2 y d2 y

Hence Eq. (6) becomes

= 41^, = X 2

dx dt dx2 dt2

t d y r- dy t —^ X —T2 + vX —— + X —— y = 0,

VX dt2 dt *Jx

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

d2y dy

+ -Ã + ty = 0. (8)

dt2 dt

6The substitution t = s/X. x 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 Bessel’s equation of order zero (see Section 5.8). The general solution of Eq. (8) for t > 0 is

y = c1 J0(t) + c2Y0(t);

hence the general solution of Eq. (7) for x > 0 is

y = q J0(VI x) + c2Y0(Vx x), (9)

where J0 and Y0 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)mxmx2m

J0(^ x) = 1 + ? ( - > - , x > 0, (10)

m=1 22m (m!)2 ’

Y0 (VX x) = -n

f VXx\ ã- Þ (-1Ã+1 HmXmx2m

Y + l.>— x) + ? 22ra (ml)2

x > 0, (11)

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

m m^(X m

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

Suppose that we seek a solution ofEq. (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) ^ - ñþ as x ^ 0, the condition y (0) = 0 can be satisfied only by choosing

c1 = 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 Y0. The second boundary condition, y(1) = 0, then yields

UVX) = 0. (15)

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

which yield the eigenvalues 0 < < X2 < ¦¦ < Xn < ¦¦ ofthe given problem. The

corresponding eigenfunctions are

ôï(x) = , (16)

7The function J is well tabulated; the roots ofEq. (15) can be found in various tables, for example, those in Jahnke and Emde or Abramowitz and Stegun. The first three roots ofEq. (15) are ë/X = 2.405, 5.520, and 8.654, respectively, to four significant figures; y'X = (n - 1/4)n for large n.

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