Answer to Question #223826 in Chemical Engineering for Lokika

Question

L^{-1} 1/s^{2}+4s+4 }

Answer 1

$L^{-1}\left\{\frac{1}{s^2+4s+4}\right\}\\ \frac{1}{s^2+4s+4}=\frac{1}{\left(s+2\right)^2}\\ =L^{-1}\left\{\frac{1}{\left(s+2\right)^2}\right\}\\ \mathrm{Apply\:inverse\:transform\:rule:\quad if\:}L^{-1}\left\{F\left(s\right)\right\}=f\left(t\right)\mathrm{\:then}\:L^{-1}\left\{F\left(s-a\right)\right\}=e^{at}f\left(t\right)\\ \mathrm{For\:}\frac{1}{\left(s+2\right)^2}:\quad a=-2,\:\quad F\left(s\right)=\frac{1}{s^2}\\ =e^{-2t}L^{-1}\left\{\frac{1}{s^2}\right\}\\ =e^{-2t}t$