Am I using the Laplace Transform pair correctly? e^t sin 2t to −(2)/((s−1)^2+(2)^2)

geneth1u

geneth1u

Answered question

2022-09-09

Laplace Transform which is as follows:
e t s i n 2 t
for
t 0
Now I know that by using the transform pair we get:
[ e a t s i n ω 0 t ] u ( t ) ω 0 ( s + α ) 2 + ( ω 0 ) 2
Using this pair we:
[ e t s i n 2 t ] u ( t ) 2 ( s + 1 ) 2 + ( 2 ) 2
Now for u ( t ) what I think should come is:
2 ( s 1 ) 2 + ( 2 ) 2
Is it correct?

Answer & Explanation

Medwsa1c

Medwsa1c

Beginner2022-09-10Added 17 answers

Start first at f ( t ) = s i n ( 2 t ) u ( t ), which has a Laplace transform of
2 s 2 + 2 2
then apply time-scaling property f ( α t ) = 1 | α | F ( s α ) with a=−1. Now, f(t) becomes s i n ( 2 t ) u ( t ), but this is just equal to s i n ( 2 t ) u ( t ) because s i n ( x ) is odd function.
Lastly, apply e t to f ( t ) = s i n ( 2 t ) u ( t ), and then remove the negative.
Anyway, the answer you gave is correct.

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