Page 340 - Schaum's Outline of Differential Equations

CHAP. 33] EIGENFUNCTION EXPANSIONS 323

33.10. Find an expansion for f(x) = e* in terms of the eigenfunctions of the Sturm-Liouville problem
/' + fy = 0; /(O) = 0, y(n) = 0.
From Problem 32.20, we have e n(x) = cos (n - ^)x for n = 1, 2,.... Substituting these functions and w(x) = 1,
a = 0, and b = Trinto Eq. (33.2), we obtain for the numerator:

and for the denominator:

Thus

and Eq. (33.1) becomes

By Theorem 33.1 this last equation is valid for all x in (0, n).

33.11. Find an expansion for f(x)= 1 in terms of the eigenfunctions of the Sturm-Li ouville problem
/' + ty = Q; y(0) = 0; /(I) = 0.
We can show that the eigenfunctions are e n(x) = sin (n - \)nx (n = 1, 2,...). Substituting these functions and
w(x) = 1, a = 0, b= 1 into Eq. (33.2), we obtain for the numerator:

and for the denominator:

Thus

and Eq. (33.1) becomes

By Theorem 33.1 this last equation is valid for all x in (0, 1).