What is laplacetransform t*e^{5t}sin(3t) ?

The answer to laplacetransform t*e^{5t}sin(3t) is (3(2s-10))/((s^2-10s+34)^2)

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laplacetransform t*e^{5t}sin(3t)

laplace transform

t \cdot e^{5 t} sin (3 t)

\frac{3 ( 2 s - 10 )}{( s ^{2} - 10 s + 34 ) ^{2}}

Solution steps

L {e^{5 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^{5 t} sin (3 t) : f (t) = e^{5 t} sin (3 t), k = 1

= (- 1)^{1} \frac{d}{d s} (L {e^{5 t} sin (3 t)})

L {e^{5 t} sin (3 t)} : \frac{3}{( s - 5 ) ^{2} + 9}

\frac{d}{d s} (\frac{3}{( s - 5 ) ^{2} + 9}) = - \frac{3 ( 2 s - 10 )}{( s ^{2} - 10 s + 34 ) ^{2}}

= (- 1)^{1} (- \frac{3 ( 2 s - 10 )}{( s ^{2} - 10 s + 34 ) ^{2}})

= \frac{3 ( 2 s - 10 )}{( s ^{2} - 10 s + 34 ) ^{2}}

\frac{\partial}{\partial y} (1 - x y)

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

x \to 0 lim (\frac{\frac{4}{x - 1} + 4}{x})

\frac{d}{d x} (\frac{x}{x + 2})

e^{4 x s i n (2 x)}

What is laplacetransform t*e^{5t}sin(3t) ?
The answer to laplacetransform t*e^{5t}sin(3t) is (3(2s-10))/((s^2-10s+34)^2)