Question

Find inverse Laplace transform of following:

1.F(s)=(S+1) / ((S+2)^2 (S+2S+4)^2 )

Answer 1

$L^{-1}\left\{\frac{\left(S+1\right)}{\left(\left(S+2\right)^2\left(S+2S+4\right)^2\right)}\right\}\\ =L^{-1}\left\{-\frac{1}{2\left(S+2\right)}-\frac{1}{4\left(S+2\right)^2}+\frac{3}{2\left(3S+4\right)}-\frac{3}{4\left(3S+4\right)^2}\right\}\\ \mathrm{Use\:the\:linearity\:property\:of\:Inverse\:Laplace\:Transform:}\\ \mathrm{For\:functions\:}f\left(s\right),\:g\left(s\right)\mathrm{\:and\:constants\:}a,\:b:\quad L^{-1}\left\{a\cdot f\left(s\right)+b\cdot g\left(s\right)\right\}=a\cdot L^{-1}\left\{f\left(s\right)\right\}+b\cdot L^{-1}\left\{g\left(s\right)\right\}\\ =-\frac{1}{2}e^{-2t}-\frac{e^{-2t}t}{4}+\frac{1}{2}e^{-\frac{4t}{3}}-\frac{e^{-\frac{4t}{3}}t}{12}$