I'm stuck with these Laplace problems

Question

I'm stuck on solving (4)(ii) and 4(b), I have tried to solve them but got stuck in the middle, any hints?

This is where I have gotten in (4)(a)(ii) and (4)(b)(i), I have no idea how to start solving on (4)(b)(ii)

Answer 1

For 4aii), Remember your Laplace transform properties and notice that

$$ \frac{\mathrm{d}}{\mathrm{d}s}\frac{s}{s^{2}+4} = -\frac{s^{2}-4}{(s^{2}+4)^{2}},$$

motivating the manipulation

$$ \frac{s^{2}}{s^{2}+4} = \frac{1}{2}\left(\frac{s^{2}-4}{(s^{2}+4)^{2}} + \frac{s^{2}+4}{(s^{2}+4)^{2}}\right).$$

Alternatively, you can compute the Bromwich integral

$$ f(t) = \frac{1}{2\pi i}\lim_{R\to\infty}\int_{a-iR}^{a+iR}\frac{s^{2}}{(s^{2}+4)^{2}}\,e^{st}\,\mathrm{d}s.$$

It involves finding the residues of the second-order poles at $s = \pm 2i$ and summing them.

4bi) is similar. Notice that

$$ \frac{9}{(s^{2}+9)^{2}} = \frac{1}{2}\left(\frac{9+s^{2}}{(s^{2}+9)^{2}} + \frac{9-s^{2}}{(s^{2}+9)^{2}}\right).$$