570 CHAPTER 12 • FOURIER. $ERIES AND THE LAPLACE TRANSFORMP(-2) -8+ 3
use Q' (s) = 3s^2 + 4s + 1, calculation reveals that Q' (-
2) =
12
_
8
+
1= -1
and Q p (± i) ( .) = ±4i +^3 1. A 1. E. (12^4 ). f ( )
2
. = -
2
=i= i. pp ymg q uat1on - 4 gives t as
I ±t - ± 4t
(
P(-2) _ 2 t P (i) it P(-i) -it
ft)= Q'(- 2 )e + Q'(i)e + Q'(-i)e= -e-^2 t + (~ -i) e'^1 + (~ +i) e- tt
eit + e - u eit _ e-•t
= - e-2t + 2 + 2 2i= -e-^2 t +cost+ 2sint.
-------~EXERCISES FOR SECTION 12 .9For Exercises 1-6, use partial fractions to find the inverse Laplace transform of
Y(s).2s + 1
1. Y(8) = 8(s-1)'2. Y(s)=2s3- s
2
+4s- 6.
54
4s^2 - 6s - 12(^3) · Y (s) = s(s+2)(s- 2)'
4. Y ( 8 ) =^8
(^3) - 582 + 68 - 6
(s - 2)^4
- Y(s) = 282 +8+3.
(8 + 2)(8 - 1)^2
4-8 - y (8) = 2 4 5.
s + s+ - Use a contour integral to find the inverse Laplace transform of Y (s) = ~
4
.
s + - Use a oontour integral to find the inverse Laplace transform of Y ( s) =
s+3
(8 - 2) (8^2 + 1) ·
For Exercises 9-12, use the heaviside expansion theorem to find the inverse
Laplace transform of Y (s).3 2 3- y ( 8 ) = s + ss (^5) - -S s +.