Page 327 - Schaum's Outline of Differential Equations

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

SOLUTIONS
A boundary-value problem is solved by first obtaining the general solution to the differential equation,
using any of the appropriate methods presented heretofore, and then applying the boundary conditions to evaluate
the arbitrary constants.

Theorem 32.1. Let y^(x) and y 2(x) be two linearly independent solutions of

Nontrivial solutions (i.e., solutions not identically equal to zero) to the homogeneous boundary-
value problem (32.3) exist if and only if the determinant

equals zero.
Theorem 32.2. The nonhomogeneous boundary-value problem defined by (32.7) and (32.2) has a unique
solution if and only if the associated homogeneous problem (32.3) has only the trivial solution.
In other words, a nonhomogeneous problem has a unique solution when and only when the associated homogeneous
problem has a unique solution.

EIGENVALUE PROBLEMS
When applied to the boundary-value problem (32.4), Theorem 32.1 shows that nontrivial solutions may
exist for certain values of 'k but not for other values of 'k. Those values of 'k for which nontrivial solutions do
exist are called eigenvalues; the corresponding nontrivial solutions are called eigenfunctions.

STURM-LIOUVILLE PROBLEMS
A second-order Sturm-Liouville problem is a homogeneous boundary-value problem of the form

wherep(x),p'(x), q(x), and w(x) are continuous on [a, b], and bothp(x) and w(x) are positive on [a, b\.
Equation (32.6) can be written in standard form (32.4) by dividing through by p(x). Form (32.6), when
attainable, is preferred, because Sturm-Liouville problems have desirable features not shared by more general
eigenvalue problems. The second-order differential equation

where a 2(x) does not vanish on [a, b], is equivalent to Eq. (32.6) if and only if a' 2(x) = a^(x) (See Problem 32.15.)
This condition can always be forced by multiplying Eq. (32.8) by a suitable factor. (See Problem 32.16.)

PROPERTIES OF STURM-LIOUVILLE PROBLEMS
Property 32.1. The eigenvalues of a Sturm-Liouville problem are all real and nonnegative.
Property 32.2. The eigenvalues of a Sturm-Liouville problem can be arranged to form a strictly increasing
infinite sequence; that is, 0 < A,j < ^ 2 < ^3 < • • • Furthermore, A, n —> °° as n —> °°.