550 CHAPTER 12 • FOURIE R SERIES AND THE LAPLACE TRANSFORM
t Corollary 12. 1 If f (t), f' (t) , and f" (t) are of exponential order, then
£(!" (t)) = s^2 f (s)-sf (0)-f' (0).
s^2 + 2
•EXAMPLE 12. 13 Show that£ (cos^2 t) = s (s 2 +
4
)'Solution If we let f (t) = cos^2 t, then f (0) = 1 and f' (t) = -2sin tcost =
- sin 2t. Because£ (- sin 2t) = 2 -
2
4, Theorem 1 2.13 implies that
s +
-282 + 4 = £(!' (t)) = sC (cos
(^2) t) -1,
from which it follows that[, (cos^2 t) = ( ~
2
4) +! = ~
2
2 +2
)'
s s + s s s + 4