Question #287237

Find the inverse Laplace Transform of {1/s^3(s^2+1)}


1
Expert's answer
2022-01-14T08:05:39-0500

We shall find the inverse laplace transform analytically of

1s3(s2+1)\displaystyle\frac{1}{s^{3}(s^{2}+1)}

We shall first resolve the expression by partial fraction

1s3(s2+1)=As+Bs2+Cs3+Ds+Es2+1\displaystyle\frac{1}{s^{3}(s^{2}+1)}=\frac{A}{s}+\frac{B}{s^{2}}+\frac{C}{s^{3}}+\frac{Ds+E}{s^{2}+1}


1As2(s2+1)+Bs(s2+1)+C(s2+1)+Ds4+Es31\equiv As^{2}(s^{2}+1)+Bs(s^{2}+1)+C(s^{2}+1)+Ds^{4}+Es^{3}

It follows that

[s4][s^{4}] A+D=0A+D=0

[s3][s^{3}] B+E=0B+E=0

[s2][s^{2}] A+C=0A+C=0

[s][s] B=0    E=0B=0 \implies E=0


We have that C=1    A=1,D=1C=1\implies A=-1, D=1


NOW, we have

1s3(s2+1)=1s+1s3+ss2+1\displaystyle\frac{1}{s^{3}(s^{2}+1)}=\frac{-1}{s}+\frac{1}{s^{3}}+\frac{s}{s^{2}+1}


L1{1s3(s2+1}=L1{1s+1s3+ss2+1}\mathcal{L}^{-1}\left\{\displaystyle\frac{1}{s^{3}(s^{2}+1}\right\}=\mathcal{L}^{-1}\left\{\frac{-1}{s}+\frac{1}{s^{3}}+\frac{s}{s^{2}+1}\right\}


=L1{1s}+L1{1s3}+L1{ss2+1}=\mathcal{L}^{-1}\left\{\frac{-1}{s}\right\}+\mathcal{L}^{-1}\left\{\frac{1}{s^{3}}\right\}+\mathcal{L}^{-1}\left\{\frac{s}{s^{2}+1}\right\}


=1+t22+cos(t)=-1+\frac{t^{2}}{2}+cos(t)

=cos(t)+t221=cos(t)+\frac{t^{2}}{2}-1

which is the inverse laplace transform of 1s3(s2+1)\displaystyle\frac{1}{s^{3}(s^{2}+1)}


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