What is the Laplace transform of $\dfrac{y'(t)}{t}$?

Question

I know that

${\mathcal{L}}\left\{ {\dfrac{1}{t}y\left( t \right)} \right\} = \int\limits_s^\infty {Y\left( u \right){\text{ d}}} u$

and that

${\mathcal{L}}\left\{ y'(t) \right\}=sY(s)-y(0)$

How can I find

${\mathcal{L}}\left\{ \dfrac{y'(t)}{t} \right\}$

Any help will be greatly appreciated

Thank you

Answer 1

As @Patrick suggested, I think the only way in which you can have a certain formula is to compose two formulas together. In fact:

$$\mathcal{L}\{y(t)/t\}=\int_s^{\infty} Y(u)du,~~Y(s)=\mathcal{L}\{y(t)\},~~\mathcal{L}\{y'(t)\}=sY(s)-y(0)$$ So:

$$\mathcal{L}\{y'(t)/t\}=\int_s^{\infty} \mathcal{L}\{y'(t)\}du=\int_s^{\infty} \big(uY(u)-y(0)\big)du$$