What is laplacetransform x^2e^{2x} ?

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

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laplacetransform x^2e^{2x}

laplace transform

x^{2} e^{2 x}

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

Solution steps

L {e^{2 x} x^{2}}

Use Laplace Transform table:

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

For

x^{2} e^{2 x} : f (x) = e^{2 x}, k = 2

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

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

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

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

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

\frac{d}{d x} (2 sec^{2} (2 x))

5 (2 x cos (3 x) - 3 x^{2} sin (3 x))

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