`3/(s^2(s+2))` `=A/s+B/s^2+C/(s+2)`

`3=As(s+2)+B(s+2)+Cs^2`

Substituting convenient values of `s` gives us:

`s=0` gives `3=2B`, which gives `B=3/2`.

`s=-2` gives `3=4C`, which gives `C=3/4`.

`s=1` gives `3=3A+3B+C`, which gives `A=-3/4`.

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

The inverse Laplace Transform is therefore:

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

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

`=-3/4+3/2t+3/4e^(-2t)`

Here is the graph of the inverse Laplace Transform function.

0.511.522.533.541234567tf(t)Open image in a new page

Graph of `f(t)==-3/4+3/2t+3/4e^(-2t)`.