We shall find the inverse laplace transform analytically of
s3(s2+1)1
We shall first resolve the expression by partial fraction
s3(s2+1)1=sA+s2B+s3C+s2+1Ds+E
1≡As2(s2+1)+Bs(s2+1)+C(s2+1)+Ds4+Es3
It follows that
[s4] A+D=0
[s3] B+E=0
[s2] A+C=0
[s] B=0⟹E=0
We have that C=1⟹A=−1,D=1
NOW, we have
s3(s2+1)1=s−1+s31+s2+1s
L−1{s3(s2+11}=L−1{s−1+s31+s2+1s}
=L−1{s−1}+L−1{s31}+L−1{s2+1s}
=−1+2t2+cos(t)
=cos(t)+2t2−1
which is the inverse laplace transform of s3(s2+1)1
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