What is inverse oflaplace (s-1)/((s+1)(s+2)s) ?

The answer to inverse oflaplace (s-1)/((s+1)(s+2)s) is 2e^{-t}-3/2 e^{-2t}-1/2 \H(t)

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inverse oflaplace (s-1)/((s+1)(s+2)s)

inverse laplace

\frac{s - 1}{( s + 1 ) ( s + 2 ) s}

2 e^{- t} - \frac{3}{2} e^{- 2 t} - \frac{1}{2} H (t)

Solution steps

L^{- 1} {\frac{s - 1}{s ( s + 1 ) ( s + 2 )}}

Take the partial fraction of

\frac{s - 1}{( s + 1 ) ( s + 2 ) s} : \frac{2}{s + 1} - \frac{3}{2 ( s + 2 )} - \frac{1}{2 s}

= L^{- 1} {\frac{2}{s + 1} - \frac{3}{2 ( s + 2 )} - \frac{1}{2 s}}

Use the linearity property of Inverse Laplace Transform:

For functions

f (s), g (s)

and constants

a, b : L^{- 1} {a \cdot f (s) + b \cdot g (s)} = a \cdot L^{- 1} {f (s)} + b \cdot L^{- 1} {g (s)}

= L^{- 1} {\frac{2}{s + 1}} - L^{- 1} {\frac{3}{2 ( s + 2 )}} - L^{- 1} {\frac{1}{2 s}}

L^{- 1} {\frac{2}{s + 1}} : 2 e^{- t}

L^{- 1} {\frac{3}{2 ( s + 2 )}} : \frac{3}{2} e^{- 2 t}

L^{- 1} {\frac{1}{2 s}} : \frac{1}{2} H (t)

= 2 e^{- t} - \frac{3}{2} e^{- 2 t} - \frac{1}{2} H (t)

What is inverse oflaplace (s-1)/((s+1)(s+2)s) ?
The answer to inverse oflaplace (s-1)/((s+1)(s+2)s) is 2e^{-t}-3/2 e^{-2t}-1/2 \H(t)