Find Laplace Transform: (1)/(s^2) (1-e^(-st_0))^2=(4)/(s^2) e^(-st_0) sinh^2 (1/2 st_0)

Lance Liu 2022-10-25 Answered
Laplace Transform: 1 s 2 ( 1 e s t 0 ) 2 = 4 s 2 e s t 0 sinh 2 ( 1 2 s t 0 )
Verify for the Laplace transform of
F ( t ) = { t 0 t < t 0 2 t 0 t t 0 t 2 t 0 0 t > 2 t 0
L { F ( t ) } = 0 t 0 t e s t   d t + t 0 2 t 0 ( 2 t 0 t ) e s t   d t = 1 s 2 ( 1 e s t 0 ) 2 = 4 s 2 e s t 0 sinh 2 ( 1 2 s t 0 )
I was able to verify everything up to 1 s 2 ( 1 e s t 0 ) 2 , but I don't see how 1 s 2 ( 1 e s t 0 ) 2 = 4 s 2 e s t 0 sinh 2 ( 1 2 s t 0 ) ?
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Answers (1)

elulamami
Answered 2022-10-26 Author has 22 answers
( 1 e x ) 2 = 4 ( 1 e x 2 ) 2 = 4 e x ( e x / 2 e x / 2 2 ) 2 = 4 e x sinh 2 ( x 2 )
Here x = s t 0
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