[UNSOLVED]Help with Laplace Transform

So... Got my second diff eq test back today. 113/150.. Lost 31 points because it seems, contrary to my personal belief, I can't do Laplace transforms to save my life.

Therefore, I was hoping someone could help me. Here's the two functions I got very very wrong:

- $\displaystyle L\{e^t*U(t-2)\} $
- $\displaystyle L\{Y = \frac {1-e^{-2s}}{(s-3)(s^2)}\}$

Some quick notes. Y = Laplace of y,U(t-2) is that switch function that is zero when t < 2 and 1 when t>=2, and L = laplace transform

Here are my answers/ways I tried to solve them:

1. $\displaystyle L\{e^t*U(t-2)\}$

$\displaystyle L\{e^t\}*L\{U(t-2)\}$

$\displaystyle \frac {e^{-2s}}{s-1}$

It seems that is entirely wrong. I'm not sure why you can't do that and I would love some input.

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

$\displaystyle y = L\{\frac {1}{(s-3)(s^2)}\} - L\{\frac {1}{(s-3)(s^2)}\}*L\{{e^{-2s}}\}$

$\displaystyle y = L\{\frac {1}{s-3}\} * L\{\frac {1}{s^2}\} - L\{\frac {1}{s-3}\} * L\{\frac {1}{s^2}\} * L\{e^{-2s}\}$

$\displaystyle y = e^{3t} * t - e^{3(t-2)} * (t-2) * U(t-2)$

It seems that is also entirely wrong. As I said earlier. I would LOVE some help!

Regards,

Harrison Jones