What is laplacetransform t^2*e^{-4t} ?

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

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laplacetransform t^2*e^{-4t}

laplace transform

t^{2} \cdot e^{- 4 t}

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

Solution steps

L {e^{- 4 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^{- 4 t} : f (t) = e^{- 4 t}, k = 2

= (- 1)^{2} \frac{d ^{2}}{d s ^{2}} (L {e^{- 4 t}})

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

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

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

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

\int sin^{4} (x) cos^{2} (x) d x

(22, 54), (62, 27)

\frac{\partial}{\partial x} (cos (x^{2} y^{2}))

\frac{d}{d t} (sin (3 t))

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