Inverse Laplace Transform without using equations I'm stuck with a question

Magdalena Norton 2022-04-21 Answered
Inverse Laplace Transform without using equations
I'm stuck with a question about Inverse Laplace Transform here, but the use of inverse laplace transform equation is forbidden.
Y(s)=eπss[(s+1)2+1]
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Answers (1)

dagars5nx
Answered 2022-04-22 Author has 12 answers

The inverse Laplace transform of 1s2+1 is sint. Therefore we have
L-1{1(s+1)2+1}=e-tsin(t)=12i(e(-1+i)t-e(-1-i)t)
Therefore
L-1{1s1(s+1)2+1}=0te-usin(u)du=12i(e(-1+i)t-1-1+i-e(-1-i)t-1-1-i)
=1212et(sin(t)+cos(t))
The last step is to use the shift theorem L1{easF(s)}=f(ta)Θ(ta) and we get
L-1{e-πs1s1(s+1)2+1}=[12-12e-(t-π)(sin(t-π)+cos(t-π))]Θ(t-π)
=(12+12eπ-t(cos(t)+sin(t)))Θ(t-π)

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