What is laplacetransform te^{6t}sin(3t) ?

The answer to laplacetransform te^{6t}sin(3t) is (3(2s-12))/((s^2-12s+45)^2)

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laplacetransform te^{6t}sin(3t)

laplace transform

t e^{6 t} sin (3 t)

\frac{3 ( 2 s - 12 )}{( s ^{2} - 12 s + 45 ) ^{2}}

Solution steps

L {e^{6 t} t sin (3 t)}

Use Laplace Transform table:

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

For

t e^{6 t} sin (3 t) : f (t) = e^{6 t} sin (3 t), k = 1

= (- 1)^{1} \frac{d}{d s} (L {e^{6 t} sin (3 t)})

L {e^{6 t} sin (3 t)} : \frac{3}{( s - 6 ) ^{2} + 9}

\frac{d}{d s} (\frac{3}{( s - 6 ) ^{2} + 9}) = - \frac{3 ( 2 s - 12 )}{( s ^{2} - 12 s + 45 ) ^{2}}

= (- 1)^{1} (- \frac{3 ( 2 s - 12 )}{( s ^{2} - 12 s + 45 ) ^{2}})

= \frac{3 ( 2 s - 12 )}{( s ^{2} - 12 s + 45 ) ^{2}}

What is laplacetransform te^{6t}sin(3t) ?
The answer to laplacetransform te^{6t}sin(3t) is (3(2s-12))/((s^2-12s+45)^2)