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Question Number 129241 by BHOOPENDRA last updated on 14/Jan/21

$$\frac{e^{-\pi s}}{s^2(s^2+1)}findinverselaplace$$

Answered by Olaf last updated on 14/Jan/21

$$\mathcal{L}\left(f\left(t\right)\right)\left(s\right)=\frac{e^{-\pi s}}{s^2(s^2+1)}=\mathrm{F}\left(s\right)$$$$\mathrm{F}\left(s\right)=\frac{e^{-\pi s}}{s^2}-\frac{e^{-\pi s}}{s^2+1}\left(1\right)$$$$\mathrm{with}\mathcal{L}\left(g(t-\mathrm{T})\right)\left(s\right)=e^{-s\mathrm{T}}\mathrm{G}\left(s\right)$$$$\mathrm{and}\mathcal{L}\left(t\right)\left(s\right)=\frac{1}{s^2},\mathcal{L}\left(\sin t\right)\left(s\right)=\frac{1}{s^2+1}$$$$\left(1\right):\mathrm{F}\left(s\right)=\mathcal{L}(t-\pi )-\mathcal{L}\left(\sin (t-\pi )\right)$$$$\mathrm{F}\left(s\right)=\mathcal{L}(t-\pi )+\mathcal{L}\left(\sin \left(t\right)\right)$$$$\Rightarrow {\mathcal{L}}^{-1}\left(\mathrm{F}\left(s\right)\right)\left(t\right)=t-\pi +\sin t=f\left(t\right)$$

Commented by BHOOPENDRA last updated on 14/Jan/21

$$Ansisnotcrctcanyoupleaserecheck$$$$Ansis[(t-\pi )+sin(t-\pi )]\times 4(t-\pi )$$

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