What is laplacetransform t^2*e^{-3t} ?

The answer to laplacetransform t^2*e^{-3t} is 2/((s+3)^3)

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laplacetransform t^2*e^{-3t}

laplace transform

t^{2} \cdot e^{- 3 t}

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

Solution steps

L {e^{- 3 t} t^{2}}

Use Laplace Transform table:

L {t^{k} f (t)} = (- 1)^{k} \frac{d ^{k}}{d s ^{k}} (L {f (t)})

For

t^{2} e^{- 3 t} : f (t) = e^{- 3 t}, k = 2

= (- 1)^{2} \frac{d ^{2}}{d s ^{2}} (L {e^{- 3 t}})

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

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

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

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

\frac{d}{d x} (\frac{x}{10} - \frac{10}{x})

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What is laplacetransform t^2*e^{-3t} ?
The answer to laplacetransform t^2*e^{-3t} is 2/((s+3)^3)