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?
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?
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 $3$.
I think you can use the following formula. $$\mathcal{L}\{t \cdot f(t)\} = -\frac{d}{ds} F(s)$$
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.