What is laplacetransform t^2sin(t) ?

The answer to laplacetransform t^2sin(t) is -(2(-3s^2+1))/((s^2+1)^3)

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laplacetransform t^2sin(t)

laplace transform

t^{2} sin (t)

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

Solution steps

L {t^{2} sin (t)}

Use Laplace Transform table:

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

For

t^{2} sin (t) : f (t) = sin (t), k = 2

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

L {sin (t)} : \frac{1}{s ^{2} + 1}

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

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

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

What is laplacetransform t^2sin(t) ?
The answer to laplacetransform t^2sin(t) is -(2(-3s^2+1))/((s^2+1)^3)