Advanced Calculus fi..

696 Advanced Calculus, Fifth EditionThe substitutionleads to the equationsu = ~(x)e''Except for a change in notation, these equations are the same as (10.124).The problems (10.124) and (10.126) are special cases of the class of Srurm-Liouville boundary value problems. A more general case is as follows:Here the function y(x) is to be a solution of the differential equation for a 5 x 5 band is to satisfy the given boundary conditions at a and b. We further assume thatr(x), p(x), q(x) have continuous derivatives over the interval and that r(x) > 0,p(x) > 0. A value of h for which (10.127) has a solution other than y(x) r 0 iscalled a characteristic value. One could allow for complex characteristic values,but it can be shown that under the assumptions made this does not arise; hencewe restrict to real-characteristic values. For each characteristic value A there is anassociated solution y(x), called a characteristic function; the functions cy(x), wherec is a constant, are also characteristic functions. It could conceivably happen thatthere are functions besides cy(x) that have the same h; it can be shown that underthe assumptions made this cannot arise:THEOREM The characteristic values of the Sturm-Liouville problem (10.127)can be numbered to form an increasing sequence: A, c h2 < . . . < A, < . . . . The correspondingcharacteristic functions can be numbered similarly to form a sequence:yn(x); each y,(x) is determined only up to a constant multiplier. The functions y,(?c)are orthogonal with respect to the weight function p(x):The Fourier series of a function F(x) with respect to the orthogonal system{my,(x)} converges uniformly to F(x) for every function F(x) having a continuousderivative for a ( x 5 b and such that F(a) = 0, F(b) = 0.For a proof of this theorem and of theorems for more general Sturm-Liouville' boundary value problems, we refer to the books of Hellwig, Titchmarsh (1962), andKamke listed at the end of this chapter. (See also Problem 13 following Section 2.18.)Because of the theorem, we can be assured that except for minor changes inform, the statements about the wave equation and heat equation in Sections 10.7and 10.9 continue to hold when the coefficients p(x), H(x) are variable [p(x) > 0,