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

The answer to laplacetransform t^3e^{-2t} is 6/((s+2)^4)

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laplacetransform t^3e^{-2t}

laplace transform

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

\frac{6}{( s + 2 ) ^{4}}

Solution steps

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

Use Laplace Transform table:

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

For

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

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

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

\frac{d ^{3}}{d s ^{3}} (\frac{1}{s + 2}) = - \frac{6}{( s + 2 ) ^{4}}

= (- 1)^{3} (- \frac{6}{( s + 2 ) ^{4}})

= \frac{6}{( s + 2 ) ^{4}}

What is laplacetransform t^3e^{-2t} ?
The answer to laplacetransform t^3e^{-2t} is 6/((s+2)^4)