Find the inverse Laplace transform of X(S)=(2+2se^(-2s)+4e^(-4s))/(s^2+4s+3) with R(s)>−1

Jaylen Dudley

Jaylen Dudley

Answered question

2022-09-09

Find the inverse Laplace transform of X ( S ) = 2 + 2 s e 2 s + 4 e 4 s s 2 + 4 s + 3 with ( s ) > 1
I am aware that this is a step function so therefore it will involve this property:
e c s F ( s ) = f ( t c ) u ( t c )
So far I factored the denominator and separated the numerator and have:
2 ( s + 1 ) ( s + 3 ) + 2 s e 2 s ( s + 1 ) ( s + 3 ) + e 4 s 4 ( s + 1 ) ( s + 3 )

Answer & Explanation

Yasmin Lam

Yasmin Lam

Beginner2022-09-10Added 13 answers

Hint: Rewrite your last equation as
2 ( s + 1 ) ( s + 3 ) + e 2 s 2 s ( s + 1 ) ( s + 3 ) + e 4 s 4 ( s + 1 ) ( s + 3 )
and then do partial fraction decomposition on all three terms. This will form
( 1 s + 1 1 s + 3 ) + e 2 s ( 3 s + 3 1 s + 1 ) + e 4 s ( 2 s + 1 2 s + 3 )
from which you can now use the property e c s F ( s ) = L { f ( t c ) u ( t c ) } ( s )

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