Did I Inverse this Laplace correctly? L^(-1) (4s)/((s-6)^3)

Pellagra3d 2022-10-13 Answered
L 1 4 s ( s 6 ) 3
4 L 1 s s 3 | s = s 6
4 L 1 1 s 2 | s = s 6
4 L 1 1 ! s 1 + 1 | s = s 6
4 t e 6 t
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Answers (2)

Phillip Fletcher
Answered 2022-10-14 Author has 21 answers
If you don't know residues, you can rewrite your function as
4 s ( s 6 ) 3 = 4 ( s 6 ) ( s 6 ) 3 + 24 ( s 6 ) 3 = 4 ( s 6 ) 2 + 24 ( s 6 ) 3
and use the "translation rule" to invert the two terms separately.
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Christopher Saunders
Answered 2022-10-15 Author has 6 answers
The ILT may be computed via the residue theorem as follows:
4 2 ! d 2 d s 2 [ s e s t ] s = 6 = 2 d d s [ ( 1 + s t ) e s t ] s = 6 = 2 [ ( t ( 1 + s t ) + t ) e s t ] s = 6 = ( 4 t + 12 t 2 ) e 6 t
You seem to be missing the t 2 term.
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