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

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y = ô(ë) = f G(x, s, ë) f (s) ds. (iii)

0

Note that in this problem the Green’s function also depends on the parameter ë.

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Chapter 11. Boundary Value Problems and Sturm Liouville Theory

(a) Show that if these two expressions for ô (x) are to be equivalent, then

^ ^ ^ ô1 (x)ô (s) ÷

G(x, s, ä) = Ó —--------------, (iv)

Ò¥1 x ³ - ä

where X i and ô t are the eigenvalues and eigenfunctions, respectively, of Eqs. (3), (2) of the text. Again we see from Eq. (iv) that ä cannot be equal to any eigenvalue X i.

(b) Derive Eq. (iv) directly by assuming that G(x, s, ä) has the eigenfunction expansion

TO

G(x, s,v) = Y a ³(x, ä)ô i (s); (v)

i=1

determine at (x, ä) by multiplying Eq. (v) by r(s)ôj(s) and integrating with respect to s from s = 0 to s = 1.

Hint: Show first that X i and ô t satisfy the equation

Ô (x) = (X . - ä) G(x, s, ä)ã(s)ô (s) ds. (vi)

J0

36. Consider the boundary value problem

-d2y/ds2 = S(s - x), y(0) = 0, y(1) = 0,

where s is the independent variable, s = x is a definite point in the interval 0 < s < 1, and

S is the Dirac delta function (see Section 6.5). Show that the solution of this problem is the Green’s function G(x, s) obtained in Problem 28.

In solving the given problem, note that S(s - x) = 0 in the intervals 0 < s < x and x < s < 1. Note further that -dy/ds experiences a jump of magnitude 1 as s passes through the value x.

This problem illustrates a general property, namely, that the Green’s function G(x, s) can be identified as the response at the point s to a unit impulse at the point x. A more general nonhomogeneous term f on 0 < x < 1 can be regarded as a continuous distribution of impulses with magnitude f (x) at the point x. The solution of a nonhomogeneous boundary value problem in terms of a Green’s function integral can then be interpreted as the result of superposing the responses to the set of impulses represented by the nonhomogeneous term f (x).

11.4 Singular Sturm-Liouville Problems

In the preceding sections of this chapter we considered Sturm-Liouville boundary value problems: the differential equation

L[y] = -[p(x)/]' + q(x)y = Xr(x)y, 0 < x < 1, (1)

together with boundary conditions of the form

a1 y(0) + a2 y (0) = 0, (2)

b, y(1) + V (1) = 0. (3)

Until now, we have always assumed that the problem is regular; that is, p is differentiable, q and r are continuous, and p(x) > 0 and r(x) > 0 at all points in the closed

11.4 Singular Sturm-Liouville Problems

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interval. However, there are also equations of physical interest in which some of these conditions are not satisfied.

For example, suppose that we wish to study Bessel’s equation of order v on the interval 0 < x < 1. This equation is sometimes written in the form6

v2

— (x/)' + — y = Xxy, (4)

x

so that p(x) = x, q(x) = v2/x, and r(x) = x. Thus p(0) = 0, r(0) = 0, and q(x) is unbounded and hence discontinuous as x ^ 0. However, the conditions imposed on regular Sturm-Liouville problems are met elsewhere in the interval.

Similarly, for Legendre’s equation we have

— [(1 — x2)/]' = Xy, — 1 < x < 1, (5)

where X = à(à + 1), p(x) = 1 — x2, q(x) = 0, and r(x) = 1. Here the required conditions on p, q, and r are satisfied in the interval 0 — x < 1 except at x = 1 where p

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