Help with unit step function and laplace transform

I could not figure out how to get the LaTex to look right with piece wise functions, so I just made a picture. Here are the 2 problems:

http://i909.photobucket.com/albums/a...ah/problem.jpg

Its sopposed to be a piece wise function but I cannot figure out do it with latex.

For no. 1 How do I set it up? Once I get past that, Im sure I can do the rest. Heres what I have:

$\displaystyle 3U(t) - e^4^t^-^1^1U(t-3)$ Is this correct?

For problem no.2, I just need to get to the undertermined coefficient part. So heres what I do for the problem:

$\displaystyle s^2Y(s) - sy(0) - y'(0) - 3sY(s) - y(0) = L{sin(t-2\pi ) U(t-2\pi )}$

Solving for $\displaystyle Y(s)$

$\displaystyle Y(s) =\frac{ e^-^2^\pi^s}{s(s^2+ 1)(s-3)} + \frac{1}{s(s-3)}$

$\displaystyle Setting up Undertermined Coefficients:$

$\displaystyle e^2^\pi ^s = \frac{As+B}{s^2+1} + \frac{C}{s} + \frac{D}{s-3}$

Does this look correct so far? Thanks guys!