find the inverse laplace transformation of (13s^2+3s+6)/((s-2)(s^2+9))

Parker Pitts

Parker Pitts

Answered question

2022-10-10

find the inverse laplace transformation of
13 s 2 + 3 s + 6 ( s 2 ) ( s 2 + 9 ) .
Here is my work:
Using partial factions:
Y ( s ) = 13 s 2 + 3 s + 6 ( s 2 ) ( s 2 + 9 ) = 64 13 ( 2 + s ) + 3 ( 83 + 35 s ) 13 ( 9 + s 2 )
And:
3 ( 83 + 35 s ) 13 ( 9 + s 2 ) = 83 9 + s 2 + 35 s 9 + s 2

Answer & Explanation

Jayleen Copeland

Jayleen Copeland

Beginner2022-10-11Added 4 answers

The partial fraction expansion yields:
3 ( 35 s + 83 ) 13 ( s 2 + 9 ) + 64 13 ( s 2 ) = 3 ( 35 s ) 13 ( s 2 + 9 ) + 3 ( 83 ) 13 ( s 2 + 9 ) + 64 13 ( s 2 )
Now, we put that result into the desired forms:
3 ( 35 s ) 13 ( s 2 + 3 2 ) + 3 ( 83 ) 13 ( s 2 + 3 2 ) + 64 13 ( s 2 )
From this, we can see the forms we need.
This yields:
y ( t ) = 1 13 ( 105 cos 3 t + 83 sin 3 t + 64 e 2 t )

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