2.9.5.7
Applying the Laplace transform yields
s
2
Y (s);sy(0);y
[1]
(0)+ 2[sY (s);y(0)]+ 5Y (s)=0:
Rearranging yields
(s
2
+2s +5)Y (s)=sy(0)+ 2y(0)+ y
[1]
(0);;
or
Y (s)=
sy(0)+ 2y(0)+ y
[1]
(0)
(s +1;j2)(s+1+j2)
For the given initial conditions we then have
Y (s) =
2s
(s +1;j2)(s +1+j2)
=
M
s +1;j2
+
M
(s +1+j2)
:
M = [(s +1;j2)Y (s)]
s=;1+j2
=
2s
(s +1+j2)
s=;1+j2
=
;2+j4
j4
=
p
5=2
6
tan
;1
(0:5) r.
y(t)=
p
5e
;t
cos[2t +0:4636]
1
