Page 334 - Schaum's Outline of Differential Equations

CHAP. 32] SECOND-ORDER BOUNDARY-VALUE PROBLEMS 317

From Eq. (29.9) we conclude that for i + k,

But since y^(x) is a nonzero function and w(x) is positive on [a, b], it follows that

hence, c k = 0. Since c k=0, k= 1, 2, ... ,p, is the only solution to (_/), the given set of functions is linearly independent
on [a, b].

Supplementary Problems

In Problems 32.22 through 32.29, find all solutions, if solutions exist, to the given boundary-value problems.

In Problems 32.30 through 32.36, find the eigenvalues and eigenfunctions, if any, of the given boundary-value problems.

In Problems 32.37 through 32.43, determine whether each of the given differential equations with the boundary conditions
X-l) + 2/(-l) = 0, XI) + 2/(l) = 0 is a Sturm-Liouville problem.

32.44. Transform e^y" + e^y' + (x + A,)y = 0 into Eq. (32.6) by means of the procedure outlined in Problem 32.16.

2
32.45. Transform x y" + xy' + Xxy = 0 into Eq. (32.6) by means of the procedure outlined in Problem 32.16.
32.46. Verify Properties 32.1 through 32.4 for the Sturm-Liouville problem

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