Find the inverse Laplace transform of the giveb function by using the convolution theorem. F(x)=(s)/((s+1)(s^2+4))

apopiw83

apopiw83

Answered question

2022-11-18

Find the inverse Laplace transform of the giveb function by using the convolution theorem.
F ( x ) = s ( s + 1 ) ( s 2 + 4 )
If I use partial fractions I get:
s + 4 5 ( s 2 + 4 ) 1 5 ( x + 1 )
which gives me Laplace inverses:
1 5 ( cos 2 t + sin 2 t ) 1 5 e t
But the answer is:
f ( t ) = 0 t e ( t τ ) cos ( 2 τ ) d τ
How did they get that?

Answer & Explanation

metodikkf6z

metodikkf6z

Beginner2022-11-19Added 14 answers

Using the fact about the Laplace transform L that
L ( f g ) = L ( f ) L ( g ) = F ( s ) G ( s ) ( f g ) ( t ) = L 1 ( F ( s ) G ( s ) ) .
In our case, given H ( s ) = 1 ( s + 1 ) s ( s 2 + 4 )
F ( s ) = 1 s + 1 f ( t ) = e t , G ( s ) = s s 2 + 4 g ( t ) = cos ( 2 t ) .
Now, you use the convolution as
h ( t ) = 0 t e ( t τ ) cos ( 2 τ ) d τ .

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