What is laplacetransform (t^2)*(sin(2*t)) ?

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

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laplacetransform (t^2)(sin(2t))

laplace transform

(t^{2}) \cdot (sin (2 \cdot t))

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

Solution steps

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

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

L {sin (2 t)} : \frac{2}{s ^{2} + 4}

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

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

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

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