evaluate L{(t+3) ^2 e^t}
Answers
Answer:
Step-by-step explanation:
We have:
A:
F
1
(
s
)
=
1
(
s
2
+
4
s
+
13
)
2
B:
F
2
(
s
)
=
1
(
s
2
+
4
)
(
s
+
1
)
2
The Laplace Convolution Theorem tells us that if we define the convolution of two function
f
(
t
)
and
g
(
t
)
by:
(
f
⋆
g
)
(
t
)
=
∫
t
0
f
(
t
−
x
)
g
(
x
)
d
x
Then:
L
{
(
f
⋆
g
)
(
t
)
}
=
F
(
s
)
G
(
s
)
⇔
L
−
1
{
F
(
s
)
G
(
s
)
}
=
(
f
⋆
g
)
(
t
)
We will need the following standard Laplace transform and inverses:
f
(
t
)
=
L
−
1
{
F
(
s
)
}
−−−−−−−−−−−−−−−−
F
(
s
)
=
L
{
f
(
t
)
}
−−−−−−−−−−−−−−
Notes
−−−−−
f
(
t
)
F
(
s
)
t
n
n
!
s
n
+
1
n
∈
N
e
a
t
1
s
−
a
a
constant
t
n
e
a
t
n
!
(
s
−
1
)
n
+
1
a
constant,
n
∈
N
sin
a
t
a
s
2
+
a
2
a
constant
e
a
t
sin
b
t
b
(
s
−
a
)
2
+
b
2
a
,
b
constant