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

The answer to laplacetransform t*e^{-2t} is 1/((s+2)^2)

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laplacetransform t*e^{-2t}

laplace transform

t \cdot e^{- 2 t}

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

Solution steps

L {e^{- 2 t} t}

Use Laplace Transform table:

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

For

t e^{- 2 t} : f (t) = e^{- 2 t}, k = 1

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

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

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

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

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

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What is laplacetransform t*e^{-2t} ?
The answer to laplacetransform t*e^{-2t} is 1/((s+2)^2)