How to calculate the inverse transform of this function: z=L^(-1) {-3s^3/(3s^4+16s^2+16)}

Linda Peters

Linda Peters

Answered question

2022-09-21

How to calculate the inverse transform of this function: z = L 1 { 3 s 3 / ( 3 s 4 + 16 s 2 + 16 ) }
The solution is:
z = 1 2 cos ( 2 t 3 ) 3 2 cos ( 2 t )

Answer & Explanation

Jamari Morgan

Jamari Morgan

Beginner2022-09-22Added 10 answers

Here   3 s 4 + 16 s 2 + 16 = ( s 2 + 4 ) ( 3 s 2 + 4 )   and hence by partial fraction we have,
3 s 3 3 s 4 + 16 s 2 + 16 = 3 s 2 ( s 2 + 4 ) 3 s 2 ( 3 s 2 + 4 )
So inverse Laplace transform,
L 1 { 3 s 3 3 s 4 + 16 s 2 + 16 } = L 1 { 3 s 2 ( s 2 + 4 ) } + L 1 { 3 s 2 ( 3 s 2 + 4 ) }
                                                                                        = 1 2 L 1 { 3 s 3 s 2 + 4 } 3 2 L 1 { s s 2 + 4 }
                                                                                        = 1 2 L 1 { s s 2 + 4 3 } 3 2 L 1 { s s 2 + 4 }
                                                                                        = 1 2 cos ( 2 t 3 ) 3 2 cos ( 2 t )
Note:
L = { sin ( a t ) } = a p 2 + a 2 L 1 { a p 2 + a 2 } = sin ( a t )
L = { cos ( a t ) } = p p 2 + a 2 L 1 { p p 2 + a 2 } = cos ( a t )

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