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I tried to solve the product below:

$3t\sin(6t)$

but it seems that getting the transform of each and multiply the result is not leading to a correct answer:

$\frac{3}{s^2}\frac{6}{s^2+36}$

How does one solve such transforms?

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    As you can easily check at http://en.wikipedia.org/wiki/Laplace_transform; the Laplace transform is not a multiplicative one as you noted above. Take the constant away. – 2012-06-08

3 Answers 3

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We know that if $ L(f(t))=F(s)$ so $ L(t.f(t))=-F’(s)$ in which $F’(s)=\frac{dF}{ds}$. Here you need just to derivative the second part of the last formula above with respect to $s$ and then multiply the result by $3$.

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    $+1\quad$ :^) $\quad$ – 2013-03-05
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I think you can use the following formula. $\mathcal{L}\{t \cdot f(t)\} = -\frac{d}{ds} F(s)$

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In fact, if you want to know $\mathcal{L}\left\{ t^n f(t) \right\}$, you can use this formula (which can be proven using integration by parts):

$ \mathcal{L}\left\{ t^n f(t) \right\} = (-1)^n \frac{d^n}{ds^n} \mathcal{L}\left\{f(t)\right\} $

So it kind of turn polynomials multiplying into derivatives.