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

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X1 / ã(õ)ô1(õ)ô2(õ) dx — X2 ² ô1 (õ)ã(õ)ô2(õ) dx = 0.

1 0 1 2 2 0 1 2

11.2 Sturm-Liouville Boundary Value Problems

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Because k2, r(x), and ô2(õ) are real, this equation becomes

(Ë1 — k2) f r(õ)ô1(õ)ô2(õ) dx = 0. (18)

Jo

Since by hypothesis k = k2, it follows that ô1 and ô2 must satisfy Eq. (15), and the theorem is proved.

Theorem 11.2.3 The eigenvalues ofthe Sturm-Liouville problem (1), (2) are all simple; that is, to each eigenvalue there corresponds only one linearly independent eigenfunction. Further, the eigenvalues form an infinite sequence, and can be ordered according to increasing magnitude so that

k < ^2 < k3 < ¦ ¦ ¦ < kn < ¦ ¦ ¦ .

Moreover, kn ^ro as n ^ro.

The proof of this theorem is somewhat more advanced than those of the two previous theorems, and will be omitted. However, a proof that the eigenvalues are simple is indicated in Problem 20.

Again we note that all the properties stated in Theorems 11.2.1 to 11.2.3 are exemplified by the eigenvalues kn = n2n2 and eigenfunctions ôï (x) = sin nn x of the example problem (10). Clearly, the eigenvalues are real. The eigenfunctions satisfy the orthogonality relation

² Ôò(x)Ôï(x) dx = I sin mnx sin nnx dx = 0, m = n, (19)

Jo J0

which was established in Section 10.2 by direct integration. Further, the eigenvalues can be ordered so that k1 < k2 < ¦ ¦ ¦ , and kn ^ro as n ^ro. Finally, to each eigenvalue there corresponds a single linearly independent eigenfunction.

We will now assume that the eigenvalues of the Sturm-Liouville problem (1), (2) are ordered as indicated in Theorem 11.2.3. Associated with the eigenvalue kn is a corresponding eigenfunction ôï, determined up to a multiplicative constant. It is often convenient to choose the arbitrary constant multiplying each eigenfunction so as to satisfy the condition

1

J0

r(x^2(x) dx = 1, n = 1, 2,.... (20)

Equation (20) is called a normalization condition, and eigenfunctions satisfying this condition are said to be normalized. Indeed, in this case, the eigenfunctions are said to form an orthonormal set (with respect to the weight function r) since they already satisfy the orthogonality relation (15). It is sometimes useful to combine Eqs. (15) and (20) into a single equation. To this end we introduce the symbol Smn, known as the Kronecker (1823-1891) delta and defined by

5mn = 10- ‘fm = (21)

mn 1 , if m = n.

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Chapter 11. Boundary Value Problems and Sturm Liouville Theory

Making use of the Kronecker delta, we can write Eqs. (15) and (20) as

>1

r(x)Ôm(x)<Pn(x) dx = Smn. (22)

1

0

Determine the normalized eigenfunctions of the problem (10):

Ó + Xy = 0, y(0) = 0, y(1) = 0.

The eigenvalues of this problem are Xj = n2, X2 = 4n2,... ,Xn = n2n2,..., and the corresponding eigenfunctions are kx sin n x, k2 sin2n x,..., kn sin nnx,..., respectively. In this case the weight function is r(x) = 1. To satisfy Eq. (20) we must choose k so that

1

0

(kn sin nn x)2 dx = 1

(23)

for each value of n. Since

k2nf sin2 nnx dx = k2n f \1 — 1 cos 2nnx) dx = 2ft,

J0 J0

Eq. (23) is satisfied if kn is chosen to be */2 for each value of n. Hence the normalized eigenfunctions of the given boundary value problem are

Ôï(x) = V2sinnnx, n = 1, 2, 3,

(24)

EXAMPLE

2

Determine the normalized eigenfunctions of the problem

/ + Xy = 0, y(0) = 0, /(1) + y(1) = 0. (25)

In Example 1 of Section 11.1 we found that the eigenvalues Xn satisfy the equation

si^T^+7^cos =0 (26)

and that the corresponding eigenfunctions are

Ôï (x) = kn sin yJXnx> (27)

where kn is arbitrary. We can determine kn from the normalization condition (20). Since r(x) = 1 in this problem, we have

j Ô² (x) dx = j sin2 ^JXn x dx

=t j0 (2—dx=^—

,2 14Xn — sin 24Xn gy/K — sin cos

n 4yvn n 2yXn

-,1 + cos2 = k2-------

11.2 Sturm-Liouville Boundary Value Problems

635

where in the last step we have used Eq. (26). Hence, to normalize the eigenfunctions ôï we must choose

kn = (i + cos2 Jt) ' (28)

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