Find ccL^(-1)(F(s)), if given F(s)=(2)/(s(s^2+4)).

Kaleigh Ayers 2022-09-13 Answered
Find L 1 ( F ( s ) ) , if given F ( s ) = 2 s ( s 2 + 4 )
I have tried as below.
To find inverse of Laplace transform, I want to make partial fraction as below.
2 s ( s 2 + 4 ) = A s + B s + C s 2 + 4 = ( A + B ) s 2 + C s + 4 A s ( s 2 + 4 ) .
After that, we have system of linear equation
A + B = 0 C = 0 4 A = 2.
Thus we have A=2, B=−2, and C=0. Now, substituting A,B,C and we have
2 s ( s 2 + 4 ) = 2 s + 2 s s 2 + 4 .
But the fact is
2 s ( s 2 + 4 ) 2 s + 2 s s 2 + 4 = 8 s 2 + 4 .
I'm stuck here. I can't make a partial fraction for F(s) and I can't find inverse of Laplace transform for F(s).
Anyone can give me hint to give me hint for this problem?
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Answers (2)

Gerardo Kent
Answered 2022-09-14 Author has 12 answers
You made a mistake because A 2
4 A = 2 then A = 1 2

We have step-by-step solutions for your answer!

Damon Cowan
Answered 2022-09-15 Author has 3 answers
Note that
2 s ( s 2 + 4 ) = 1 2 s s 2 ( s 2 + 4 )
So, you have
L 1 { 2 s ( s 2 + 2 ) } = 1 2 L 1 { 1 s } 1 2 L 1 { s s 2 + 4 }
Finally, you can use
L 1 { s s 2 + 4 } = cos ( 2 t )
and
L 1 { 1 s } = 1

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