4:
The Laplace transform of the function f(t) = Se-cost dt is
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S + 1
1
s 'S2 + 2s + 2
S +1
S2 + 2s + 2
1 s-1
SS2 + 2s + 2
1 S +1
s 'S2 + 2s - 2
Answers
Answer:
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In the first law, an object will not change its motion unless a force acts on it. In the second law, the force on an object is equal to its mass times its acceleration. In the third law, when two objects interact, they apply forces to each other of equal magnitude and opposite direction.
Step-by-step explanation:
L{f(t)} = F(s), then the inverse Laplace transform of F(s) is
L
−1
{F(s)} = f(t). (1)
The inverse transform L
−1
is a linear operator:
L
−1
{F(s) + G(s)} = L
−1
{F(s)} + L
−1
{G(s)}, (2)
and
L
−1
{cF(s)} = cL
−1
{F(s)}, (3)
for any constant c.
2. Example: The inverse Laplace transform of
U(s) = 1
s
3
+
6
s
2 + 4
,
is
u(t) = L
−1
{U(s)}
=
1
2
L
−1
2
s
3
+ 3L
−1
2
s
2 + 4
=
s
2
2
+ 3 sin 2t. (4)
3. Example: Suppose you want to find the inverse Laplace transform x(t) of
X(s) = 1
(s + 1)4
+
s − 3
(s − 3)2 + 6
.
Just use the shift property (paragraph 11 from the previous set of notes):
x(t) = L
−1
1
(s + 1)4
+ L
−1
s − 3
(s − 3)2 + 6
=
e
−t
t
3
6
+ e
3t
cos √
6t.
4. Example: Let y(t) be the inverse Laplace transform of
Y (s) = e
−3s
s
s
2 + 4