What is laplacetransform t*cos(3t) ?

The answer to laplacetransform t*cos(3t) is -(-s^2+9)/((s^2+9)^2)

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laplacetransform t*cos(3t)

laplace transform

t \cdot cos (3 t)

- \frac{- s ^{2} + 9}{( s ^{2} + 9 ) ^{2}}

Solution steps

L {t cos (3 t)}

Use Laplace Transform table:

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

For

t cos (3 t) : f (t) = cos (3 t), k = 1

= (- 1)^{1} \frac{d}{d s} (L {cos (3 t)})

L {cos (3 t)} : \frac{s}{s ^{2} + 9}

\frac{d}{d s} (\frac{s}{s ^{2} + 9}) = \frac{- s ^{2} + 9}{( s ^{2} + 9 ) ^{2}}

= (- 1)^{1} \frac{- s ^{2} + 9}{( s ^{2} + 9 ) ^{2}}

= - \frac{- s ^{2} + 9}{( s ^{2} + 9 ) ^{2}}

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What is laplacetransform t*cos(3t) ?
The answer to laplacetransform t*cos(3t) is -(-s^2+9)/((s^2+9)^2)