What is inverse oflaplace {((s+2)^2)/(s^3)} ?

The answer to inverse oflaplace {((s+2)^2)/(s^3)} is \H(t)+4t+2t^2

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inverse oflaplace {((s+2)^2)/(s^3)}

inverse laplace

{\frac{( s + 2 ) ^{2}}{s ^{3}}}

H (t) + 4 t + 2 t^{2}

Solution steps

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

Expand

\frac{( s + 2 ) ^{2}}{s ^{3}} : \frac{1}{s} + \frac{4}{s ^{2}} + \frac{4}{s ^{3}}

= L^{- 1} {\frac{1}{s} + \frac{4}{s ^{2}} + \frac{4}{s ^{3}}}

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{1}{s}} + L^{- 1} {\frac{4}{s ^{2}}} + L^{- 1} {\frac{4}{s ^{3}}}

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

L^{- 1} {\frac{4}{s ^{2}}} : 4 t

L^{- 1} {\frac{4}{s ^{3}}} : 2 t^{2}

= H (t) + 4 t + 2 t^{2}

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What is inverse oflaplace {((s+2)^2)/(s^3)} ?
The answer to inverse oflaplace {((s+2)^2)/(s^3)} is \H(t)+4t+2t^2