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

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

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laplacetransform t^2e^{at}

laplace transform

t^{2} \cdot e^{a \cdot t}

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

Solution steps

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

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

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

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

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

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

t^{2} y^{^{''}} + 19 t y^{^{'}} + 81 y = 0

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\frac{\partial}{\partial x} (ln (8 x^{2} + 3 y^{2} + 3))

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