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s2
3
+4
3
3
2
3
=
sin(2t)
s2 + 4
2 s2 + 2 2
2
3s + 1
s6
3s + 1
A
B
3s + 1
=
=
+
2
s s6
(s 3)(s + 2)
s3 s+2
A = 7/5, B = 3/5
7/5
3/5
7/5e3t ,
3/5e2t
s3
s+2
3/5
f (t) =
+ 3/5e2t
s+2
(b) F (s) =
(c) F (s) =
s2
s2
2s + 1
s2 2s + 2
2s + 1
2s + 1
2(s 1) + 3
=
=
2 cos t et + 3 sin t et
2
2s + 2
(s 1) + 1
(s 1)2 + 1
(d) F (s) =
8s2 4s + 12
s(s2 + 4)
2s2 s + 3
8s2 4s + 12
=
4
s(s2 + 4)
s(s2 + 4)
2s2 s + 3
A Bs + C
=
+ 2
s(s2 + 4)
s
s +4
A = 3/4, B = 5/4, C = 1
8s2 4s + 12
A Bs + C
3 5s 4
= + 2
=4
+ 2
2
s(s + 4)
s
s +4
s s +4
3
3, or 3u(t)
s
5s 4
5 cos(2t) 2 sin(2t)
s2 + 4
f (t) = 3 + 5 cos(2t) 2 sin(2t) or 3u(t) + 5 cos(2t) 2 sin(2t)
1
y(0) = 1, y 0 (0) = 0
Method 1:
s+3
s+3
A
B
=
=
+
+ 3s + 2
(s + 2)(s + 1)
s+2 s+1
A = 1, B = 2
1
e2t
s+2
2
2et
s+1
y(t) = e2t + 2et
Y =
s2
Method 2:
s+3
s+3
=
+ 3s + 2
(s + 3/2)2 (1/2)2
s + 3/2
3/2
=
+
2
2
(s + 3/2) (1/2)
(s + 3/2)2 (1/2)2
1/2t
s + 3/2
+ e1/2t
et + e2t
3/2t
3/2t e
e
cosh(1/2t)
=
e
=
(s + 3/2)2 (1/2)2
2
2
1/2t
e1/2t
3et 3e2t
3/2
3/2t
3/2t e
3e
sinh(1/2t)
=
3e
=
(s + 3/2)2 (1/2)2
2
2
t
2t
t
2t
e +e
3e 3e
y(t) =
+
= e2t + 2et
2
2
Y =
s2
(b) y 00 4y 0 + 4y = 0;
y(0) = 1, y 0 (0) = 1
s2 Y s 1 4[sY 1] + 4Y = 0
(s2 4s + 4)Y = s 3
s3
s3
1
(s 2) 1
1
=
=
=
Y = 2
2
2
s 4s + 4
(s 2)
(s 2)
s 2 (s 2)2
1
e2t
s2
1
t e2t
(s 2)2
y(t) = e2t te2t
(c) y 00 2y 0 + 2y = cos t;
y(0) = 1, y 0 (0) = 0
s2 Y s 2[sY 1] + 2Y =
s
s2 + 1
s
+s2
+1
s2
s
+ 2
Y = 2
2
(s + 1)(s 2s + 2) s 2s + 2
s
As + B
Bs + D
= 2
+ 2
2
2
(s + 1)(s 2s + 2)
s +1
s 2s + 2
A = 1/5, B = 2/5, C = 1/5, D = 4/5
s
1 s2
s + 4
=
+
(s2 + 1)(s2 2s + 2)
5 s2 + 1 s2 2s + 2
s
2
s2
= 2
+ 2
cos t 2 sin t
2
s +1
s +1 s +1
s + 4
(s 1) + 3
=
cos t et + 3 sin t et
2
s 2s + 2
(s 1)2 + 1
(s2 2s + 2)Y =
y=
s2
1
[cos t 2 sin t et cos t + 3et sin t]
5
(d) y 00 2y 0 + 2y = et ;
y(0) = 0, y 0 (0) = 1
s2 Y 1 2sY + 2Y =
1
s+1
s+2
1
+1=
s+1
s+1
A
Bs + C
s+2
=
+ 2
Y =
2
(s + 1)(s 2s + 2)
s + 1 s 2s + 2
A = 1/5 B = 1/5 C = 8/5
1
1
s + 8
Y =
+
5 s + 1 s2 2s + 2
1
et
s+1
s + 8
(s 1) + 7
=
cos t et + 7 sin t et
2
s 2s + 2
(s 1)2 + 1
(s2 2s + 2)Y =
y=
1 t
[e et cos t + 7et sin t]
5