Determine the Laplace transform of the following functions. \frac{2}{t}\sin 3at where a is any positive constant and s>a

Chesley 2021-09-08 Answered
Determine the Laplace transform of the following functions.
2tsin3at where a is any positive constant and s>a
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Expert Answer

firmablogF
Answered 2021-09-09 Author has 92 answers
Step 1
Given:
The function is 2tsin(3at), where a is any positive constant and s>a
Step 2
Let the function
f(t)=2tsin(3at)
if L[f(t)]=F(s),then  L[1tf(t)]=sF(s)ds
We know that
L[sin(at)]=as2+a2
Then
L[sin(3at)]=3as2+(3a)2
L[sin(3at)]=3as2+9a2
Step 3
Then
L[2tsin(3at)]=2L[1tsin(3at)]
Since L[1tf(t)]=sF(s)ds
2L[1tsin(3at)]=203as2+9a2ds
L[2tsin(3at)]=6a01s2+(3a)2ds
L[2tsin(3at)]=6a[13atan1(13a)]0
L[2tsin(3at)]=2[π20]
Therefore, L[2tsin(3at)]=π
Step 4
Answer:
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