Laplace transform of following function, is this correct? 4H(t−3) * t^2

Kamden Larson 2022-10-24 Answered
Laplace transform (Using tables) of following function, is this correct?
4 H ( t 3 )   . t 2
For my answer I used the the t-shift theorem (using tables) and I got
4 e 3 s . 2 s 3
I have a feeling this isn't correct.
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Answers (1)

Zackary Mack
Answered 2022-10-25 Author has 12 answers
We need to write t 2 in terms of t−3 to use the Shifting Property or Laplace Transform definition (integral instead of tables).
We have
t 2 = ( ( t 3 ) + 3 ) 2 = ( t 3 ) 2 + 6 ( t 3 ) + 9
Hence, we can write
4 H ( t 3 ) t 2 = 4 H ( t 3 ) ( ( t 3 ) 2 + 6 ( t 3 ) + 9 )
Using those three pieces, we end up with
L (   4   H ( t 3 )   t 2 ) = 4 e 3 s ( 9 s 2 + 6 s + 2 ) s 3
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