`G(s)=(2s^2-16)/(s^3-16s)`

`=(2s^2-16)/(s(s^2-16))`

`=(2s^2-16)/(s(s+4)(s-4))`

`=A/s+B/(s+4)+C/(s-4)`

Multiplying throughout by `s^3-16s` gives:

`2s^2-16` `=A(s^2-16)+` `Bs(s-4)+` `Cs(s+4)`

Substituting `s=4` gives `16=32C`, which gives us `C=1/2`.

`s=-4` gives `16=32B`, which gives `B=1/2`.

`s=0` gives `-16=-16A`, which gives `A=1`.

So `(2s^2-16)/(s^3-16s)` `=1/s+1/(2(s-4))+1/(2(s+4))`

So the inverse Laplace Transform is given by:

`Lap^{:-1:}{(2s^2-16)/(s^3-16s)} =Lap^{:-1:}{1/s+1/(2(s-4))+1/(2(s+4))}`

`=1+1/2e^(4t)+1/2e^(-4t)`

Here is the graph of the inverse Laplace Transform function.

0.20.40.60.812345678910tf(t)Open image in a new page

Graph of `f(t)=1+1/2e^(4t)+1/2e^(-4t)`.

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