Page 331 - Schaum's Outline of Differential Equations

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

positive. To satisfy the boundary conditions, B = 0 and either A = 0 or sinv^- = 0. This last equation is
equivalent to v^- = njt where n= 1, 2, 3, .... The choice A = 0 results in the trivial solution; the choic
V^- = nn results in the nontrivial solution y n = A n sin mtx. Here the notation A n signifies that the arbitrary
constant A n can be different for different values of n.
Collecting the results of all three cases, we conclude that the eigenvalues are X n = if if and the corresponding
eigenfunctions arey n = A n sin nnx, for n= 1,2,3, ....

32.11. Find the eigenvalues and eigenfunctions of

As in Problem 32.10, the cases X = 0, X < 0, and X > 0 must be considered separately.
A, = 0: The solution is y = Cj + c 2x. Applying the boundary conditions, we obtain Cj = c 2 = 0; hence y = 0.

K<0: The solution is y = c^e + C 2e~ , where -X and •J-'X. are positive. Applying the boundary conditions,
we obtain

which admits only the solution c l = c 2 = 0; hence y = 0.
A,>0: The solution is y = A sin •y'kx + BcasyJix. Applying the boundary conditions, we obtain B = 0 and
A^J'k cos V^- 7f = 0. For 6 > 0, cos 6 = 0 if and only if 6 is a positive odd multiple of n!2; that is, when
9 = (2n — 1)(7T / 2) = (n — \)n, where n= 1, 2,3, .... Therefore, to satisfy the boundary conditions,

must have B = 0 and either A = 0 or cos V^- n = 0. This last equation is equivalent to v^- = n — j. Th
choice A = 0 results in the trivial solution; the choice V^- = n ~\ results in the nontrivial solution
y n = A n sin(«-|)j:.

Collecting all three cases, we conclude that the eigenvalues are 'k n = (n — -j) and the corresponding eigen-
functions are y n = A n sin (n — j")x, where n=l,2,3, ....

32.12. Show that the boundary-value problem given in Problem 32.10 is a Sturm-Liouville problem.
It has form (32.6) with p(x) = 1, q(x) = 0, and w(x) = 1. Here both p(x) and w(x) are positive and continuous
everywhere, in particular on [0, 1].

32.13. Determine whether the boundary-value problem

is a Sturm-Liouville problem.
2
x
Here/>(X) =x, q(x) =x + 1, and w(x) = e . Since bo\hp(x) and q(x) are continuous and positive on [1, 2], the
interval of interest, the boundary problem is a Sturm-Liouville problem.
32.14. Determine which of the following differential equations with the boundary conditions y(0) = 0, /(I) = 0
form Sturm-Liouville problems:

x
(a) The equation can be rewritten as (e*y')' + 'ky = 0; hence p(x) = e , q(x) = 0, and w(x) = 1. This is a
Sturm-Liouville problem.