Find inverse laplace transform : F(s)=(2 omega^3)/((s^2+omega^2)^2)

Jaiden Elliott

Jaiden Elliott

Answered question

2022-11-06

Find inverse laplace transform :
F ( s ) = 2 ω 3 ( s 2 + ω 2 ) 2

Answer & Explanation

Berattirna

Berattirna

Beginner2022-11-07Added 19 answers

We know L ( e a t ) = 1 s a
Putting a = i b , L ( e i b t ) = 1 s i b
or L ( cos b t ) + i L ( sin b t ) = s + i b s 2 + b 2
So, L ( cos b t ) = s s 2 + b 2 and L ( sin b t ) = b s 2 + b 2
Again, as L ( t n ) = n ! s n + 1
and L { e c t f ( t ) } = F ( s c ) where F ( s ) = L { f ( t ) } ,
L ( t n e i b t ) = n ! ( s i b ) n + 1 = n ! ( s + i b ) n + 1 ( s i b ) n + 1 ( s + i b ) n + 1 = n ! ( s + i b ) n + 1 ( s 2 + b 2 ) n + 1
Putting n = 1 , L ( t e i b t ) = ( s + i b ) 2 ( s 2 + b 2 ) 2 L ( t cos b t ) = s 2 b 2 ( s 2 + b 2 ) 2
Let 1 ( s 2 + b 2 ) 2 = A 1 s 2 + b 2 + B s 2 b 2 ( s 2 + b 2 ) 2
or, 1 = A ( s 2 + b 2 ) + B ( s 2 b 2 ) = s 2 ( A + B ) + b 2 ( A B )
So, A + B = 0 , b 2 ( A B ) = 1 B = A = 1 2 b 2
So, L 1 ( 1 ( s 2 + b 2 ) 2 ) = 1 2 b 2 ( sin b t b t cos b t )

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