Page 333 - Schaum's Outline of Differential Equations

316 SECOND-ORDER BOUNDARY-VALUE PROBLEMS [CHAP. 32

or
3 x
3
4
This last equation is precisely (32.6) withp(x) = (x + 2) , q(x) = (x + 2) x, and w(x) = (x + 2) e . Note, that since we
divided by a 2(x), it is necessary to restrict x 2 -2. Furthermore, in order that bothp(x) and w(x) be positive, we must
require x > -2.
32.19. Verify Properties 32.1 through 32.4 for the Sturm-Liouville problem

2 2
Using the results of Problem 32.10 we have that the eigenvalues are X n = w ^ and the corresponding eigen-
functions are y n(x) = A n sin nnx, for n = 1, 2, 3, ... The eigenvalues are obviously real and nonnegative, and they can
2
•
be ordered as ^=7? < 1^ = 47? <X 3 =9?i < ••. Each eigenvalue has a single linearly independent eigenfunction
e n(x) = sin nnx associated with it. Finally, since

we have for n ^ m and w(x) = 1:

32.20. Verify Properties 32.1 through 32.4 for the Sturm-Liouville problem

For this problem, we calculate the eigenvalues 'k n = (n — -j) and the corresponding eigenfunctions
y n (x) = A n cos (n — j)x, for n = 1, 2, .... The eigenvalues are real and positive, and can be ordered as

Each eigenvalue has only one linearly independent eigenfunction e n(x) = cos (n — j~)x associated with it. Also, for
n^m and w(x) = 1,

32.21. Prove that if the set of nonzero functions {yi(x), y2(x), . ..,y p(x)} satisfies (32.9), then the set is linearly
independent on [a, b\.
From (8.7) we consider the equation

Multiplying this equation by w(x)y k(x) and then integrating from a to b, we obtain