How to find inverse laplace of ((s^2−6s+7))/((s^2−4s+5)^2)?

Krish Crosby 2022-09-22 Answered
How to find inverse laplace of ( s 2 6 s + 7 ) ( s 2 4 s + 5 ) 2 ?
i tried partial frictions but i get 2 2 s ( s 2 4 s + 5 ) 2 + 1 s 2 4 s + 5
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Answers (1)

berzamauw
Answered 2022-09-23 Author has 9 answers
Just another way:
L s 1 [ s 2 6 s + 7 ( s 2 4 s + 5 ) 2 ] ( t ) = t ( L s 1 [ s 2 6 s + 7 ( s 2 4 s + 5 ) 2 d s ] ( t ) ) = t ( L s 1 [ 3 s 5 4 s + s 2 ] ( t ) )
Using Heaviside's expansion formula for: L s 1 [ 3 s 5 4 s + s 2 ] ( t ) we have:
P ( s ) = 3 s
Q ( s ) = 5 4 s + s 2
Q ( s ) = 4 + 2 s
solve roots for:Q(s)
α 1 = 2 i and α 2 = 2 i
P ( α 1 ) exp ( t α 1 ) Q ( α 1 ) + P ( α 2 ) exp ( t α 2 ) Q ( α 2 ) = P ( 2 i ) exp ( t ( 2 i ) ) Q ( 2 i ) + P ( 2 + i ) exp ( t ( 2 + i ) ) Q ( 2 + i ) = ( 1 2 + i 2 ) exp ( t ( 2 i ) ) + ( 1 2 i 2 ) exp ( t ( 2 + i ) ) = e 2 t ( cos ( t ) + sin ( t ) )
and then:
t ( L s 1 [ 3 s 5 4 s + s 2 ] ( t ) ) = t e 2 t ( cos ( t ) + sin ( t ) ) = e 2 t t ( cos ( t ) sin ( t ) )
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