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

ISBN 0-471-31999-6

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Moreover, kn ^to as n ^to.

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 0n(x) = sin nnx of the example problem (10). Clearly, the eigenvalues are real. The eigenfunctions satisfy the orthogonality relation

I (x)0n(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 ^ < ^2 < ? ? ? , and kn ^ to as n ^ to. 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 0n, determined up to a multiplicative constant. It is often convenient to choose the arbitrary constant multiplying each eigenfunction so as to satisfy the condition

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

Jo

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

= 10’ fm = (21)

mn 1 , if m = n.

634

Chapter 11. Boundary Value Problems and Sturm Liouville Theory

EXAMPLE

1

EXAMPLE

2

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

-1

r(x)<pm(x)<pn(x) dx = Smn. (22)

1

Jo

Determine the normalized eigenfunctions of the problem (10):

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

The eigenvalues of this problem are X1 = n2, X2 = 4n2, ...,X = n2n2,..., and

the corresponding eigenfunctions are ^ sin n x, k2 sin 2n x,..., kn sin nnx,..., respectively. In this case the weight function is r(x) = 1. To satisfy Eq. (20) we must choose kn so that

f (kn sin nnx)2 dx = 1 (23)

for each value of n. Since

k2n I sin2 nnx dx = k2n I (2 — | cos 2nnx) dx = 2k2n,

Eq. (23) is satisfied if kn is chosen to be \f2 for each value of n. Hence the normalized eigenfunctions of the given boundary value problem are

(x) = V2sinnnx, n = 1, 2, 3,.... (24)

Determine the normalized eigenfunctions of the problem

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

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

si^T^ + JXn cos fin = 0 (26)

and that the corresponding eigenfunctions are

tyn(x) = kn sin^x, (27)

where kn is arbitrary. We can determine kn from the normalization condition (20). Since

r(x) = 1 in this problem, we have

J $1 (x) dx = kl' j sin2 fin x dx

= tlfo (i - 2 cos 2 fnx) dx = t (2 -

2 - sin 2/kn - sin Jkn cosJkn

n ^ n 2^X-n

g 1 + cos2 y/K

11.2 Sturm-Liouville Boundary Value Problems

635

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

kn = (l + cos2 jr) ? (28)

The normalized eigenfunctions of the given problem are

a/2 sin a/v"x

(x) = (1 + CO^)* ; " = >• 2..... (29)

We now turn to the question of expressing a given function f as a series of eigenfunctions of the Sturm-Liouville problem (1), (2). We have already seen examples of such expansions in Sections 10.2 to 10.4. For example, it was shown there that if f is continuous and has a piecewise continuous derivative on 0 < x < 1, and satisfies the boundary conditions f (0) = f (1) = 0, then f can be expanded in a Fourier sine series of the form

TO

f(x) = E bn sin nn x. (30)

n=1

The functions sin nnx, n = 1, 2,, are precisely the eigenfunctions of the boundary value problem (10). The coefficients bn are given by

bn = 2 f f (x) sin nn x dx (31)

J0

and the series (30) converges for each x in0 < x < 1. In a similar way f can be expanded in a Fourier cosine series using the eigenfunctions cos nnx, n = 0, 1, 2,, of the boundary value problem y" + ky = 0, y (0) = 0, y (1) = 0.

Now suppose that a given function f, satisfying suitable conditions, can be expanded in an infinite series of eigenfunctions of the more general Sturm-Liouville problem

(1), (2). If this can be done, then we have

TO

f (x) = ^Z cn^n(x), (32)

n=1

where the functions $n (x) satisfy Eqs. (1), (2), and also the orthogonality condition (22). To compute the coefficients in the series (32) we multiply Eq. (32) by r(x)$m(x), where m is a fixed positive integer, and integrate from x = 0 to x = 1. Assuming that the series can be integrated term by term we obtain

/» 1 TO /» 1 TO

/ r (x) f (x)0m (x) dx =Y] cl r (x)0m (x)<pn (x) dx =Y] cn Smn. (33)

^0 n=l ^0 n=i

Hence, using the definition of Smn, we have

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