How to prove this (s)/((s^2+4)^2) is the Laplace transform of a continuous exponential f(x)?

Jadon Stein

Jadon Stein

Answered question

2022-09-07

How to prove this
s ( s 2 + 4 ) 2
is the Laplace transform of a continuous exponential f(x)?

Answer & Explanation

Peyton Cox

Peyton Cox

Beginner2022-09-08Added 18 answers

If you compare it with tables, you can discover that
f ( t ) = t sin ( 2 t ) 4 = L 1 [ s ( s 2 + 4 ) 2 ] .
To finish, take the Laplace transform of this f(t) and show that you get the RHS.
equipokypip1

equipokypip1

Beginner2022-09-09Added 5 answers

You can calculate the inverse Laplace Transform:
L { f ( t ) } = { s ( s 2 + 4 ) 2 }
L { f ( t ) } = 1 2 { 2 s ( s 2 + 4 ) 2 }
L { f ( t ) } = 1 2 { d d s 1 ( s 2 + 4 ) }
L { f ( t ) } = 1 4 { d d s 2 ( s 2 + 4 ) }
L { f ( t ) } = 1 4 { d d s L { sin ( 2 t ) } }
f ( t ) = 1 4 t sin ( 2 t )

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