Page 338 - Schaum's Outline of Differential Equations

CHAP. 33] EIGENFUNCTION EXPANSIONS 321

33.5. Find a Fourier sine series for/(X) = 1 on (0, 5).
Using Eq. (33.4) with L = 5, we have

Thus Eq. (33.3) becomes

Since/(X) = 1 is piecewise smooth on [0, 5] and continuous everywhere in the open interval (0, 5), it follows from
Theorem 33.1 that (1) is valid for all x in (0, 5).

33.6. Find a Fourier cosine series for/(X) = x on (0, 3).
Using Eq. (33.6) with L = 3, we have

Thus Eq. (33.5) becomes

Since/(X) = x is piecewise smooth on [0, 3] and continuous everywhere in the open interval (0, 3), it follows from
Theorem 33.1 that (1) is valid for all x in (0, 3).

33.7. Find a Fourier sine series for f(x) =

Using Eq. (33.4) with L = 3, we obtain