Find the Inverse Laplace transform of ccL^(-1) ((s+lambda))/(((s+lambda)^2+mu^2)(s^2+1))

dizxindlert7

dizxindlert7

Answered question

2022-09-07

Find the Inverse Laplace transform of L 1 ( s + λ ) ( ( s + λ ) 2 + μ 2 ) ( s 2 + 1 )

Answer & Explanation

acilschoincg8

acilschoincg8

Beginner2022-09-08Added 12 answers

Use convolution theorem.
x ( t ) = L 1 { s + λ ( ( s + λ ) 2 + μ 2 ) ( s 2 + 1 ) } = L 1 { s + λ ( s + λ ) 2 + μ 2 } L 1 { 1 s 2 + 1 }
x ( t ) = ( e λ t cos ( μ t ) ) sin ( t ) = 0 t e λ y cos ( μ y ) sin ( t y ) d y
You may then use the following formulas to evaluate the integral.
cos a sin b = 1 2 [ sin ( a + b ) sin ( a b ) ]
e a z sin ( b z ) d z = e a z a 2 + b 2 [ a sin ( b z ) b cos ( b z ) ]
pobi1k

pobi1k

Beginner2022-09-09Added 2 answers

hint
use this one
( s + λ ) ( ( s + λ ) 2 + μ 2 ) ( s 2 + 1 ) = A ( s + λ ) + i μ + B ( s + λ ) i μ + C s + i + D s i

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