Hi,
For Question 2,
I know you can do them by partial intergration but I'm not sure how to show them using the method it asks.
For Question 4, I genuinely have no idea.
Any help would be appreciated.
Thanks
By "using the definition" they mean
$\large F(s) = \mathscr{L}\left \{f(t) \right \} = \displaystyle{\int_{-\infty}^{\infty}}f(t)e^{-st}~dt$
and for 2(a) you would use "integration by parts" to evaluate this integral.
for 2(b) just use their hint and do the integration.
I should note that it looks like they implicitly mean f(t)=0 for t<0, otherwise these transforms won't converge.
for 4) you should have read about using Laplace transforms to solve linear constant coefficient differential equations. This is a pretty straightforward example assuming H(t) stands for the Heaviside step function. Go re-read that section of your text. Or look at this.
No, that doesn't even make sense - derivative of F(s) with respect to t?
The Laplace transform of f'(t) has an easy-to-write relationship to the Laplace transform of f(t). It should be in your textbook, or you can look at romsek's link.
- Hollywood