Find the Inverse Laplace transform of: F(s)=(1)/(s(s^2+8s+4))

Nathanael Perkins 2022-09-26 Answered
Please help me find the Inverse Laplace transform of:
F ( s ) = 1 s ( s 2 + 8 s + 4 )
After completing the square, I obtained
F ( s ) = 1 s ( ( s + 4 ) 2 12 )
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Answers (1)

ruinsraidy4
Answered 2022-09-27 Author has 17 answers
Using partial fraction decomposition gives
F ( s ) = 1 s ( s 2 + 8 s + 4 ) = 1 4 ( 1 s s s 2 + 8 s + 4 8 s 2 + 8 s + 4 )
F ( s ) = 1 4 ( 1 s s ( s + 4 ) 2 12 8 ( s + 4 ) 2 12 )
Then in order to use
L { e a t } = 1 s a
L { e a t sinh ( b t ) } = b ( s a ) 2 b 2
L { e a t cosh ( b t ) } = s a ( s a ) 2 b 2
we can rewrite F(s) as
F ( s ) = 1 4 ( 1 s s + 4 ( s + 4 ) 2 12 4 ( s + 4 ) 2 12 )
= L { 1 4 ( 1 e 4 t cosh ( 12 t ) 4 12 e 4 t sinh ( 12 t ) ) }
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