What is laplacetransform x*e^{-Ax} ?

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

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laplacetransform x*e^{-Ax}

laplace transform

x \cdot e^{- A x}

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

Solution steps

L {e^{- A x} x}

Use Laplace Transform table:

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

For

x e^{- A x} : f (x) = e^{- A x}, k = 1

= (- 1)^{1} \frac{d}{d s} (L {e^{- A x}})

L {e^{- A x}} : \frac{1}{s + A}

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

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

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

\frac{d}{d x} (\frac{x}{( 2 x - 1 ) ^{6}})

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