Math Help - Laplace of differential equation

1. Laplace of differential equation

I am having trouble with the problem attached, I need to find the inverse laplace of X(s).

I can do the 3rd term from my table of conversions directly, the 2nd term I have an answer for using partial fractions (although i'm not 100% it's correct). The 1st term is my biggest problem though.

The answer to aim for is: x(t) = 1 - e^(2t)

2. Hmm. I'm not sure I buy your derivation there. Here's the original DE:

$\ddot{x}-4\dot{x}+4x=4;\quad x(0)=0,\;\dot{x}(0)=-2.$

The LT of the DE is

$s^{2}X(s)-s\underbrace{x(0)}_{0}-\underbrace{\dot{x}(0)}_{-2}-4(sX(s)-\underbrace{x(0)}_{0})+4X(s)=\dfrac{4}{s},$ or

$s^{2}X(s)+2-4sX(s)+4X(s)=\dfrac{4}{s}.$

Solving for $X(s),$ I get

$X(s)=-\dfrac{2}{(s-2)s}.$

I think this step was incorrect:

$\mathcal{L}[4x]=\dfrac{4}{s^{2}}.$ It should be

$\mathcal{L}[4x]=4X(s).$

3. Ah yes, I see where I went wrong now, thanks!

4. You're welcome!