; x(0)=o, x'(0)=2, y(0)=0
Thank you for helping me understand how to solve this system of equations!!!
Here is the first one
Taking the transform of both sides gives
$\displaystyle s^2X-2+3sY+3Y=0$
and
$\displaystyle s^2X-2+3Y=\frac{1}{(s+1)}$
Now we have a system of Algebraic equaions. So isolate Y is the 2nd equation to get
$\displaystyle 3sY=-\frac{1}{s+1} \iff Y= -\frac{1}{3}\left( \frac{1}{s(s+1)}\right) $
Taking the inverse transform (by any method you know)
Here is using the convolution theorem we get
$\displaystyle y(t)=-\frac{1}{3} \int_{0}^{t}e^{-\tau}d\tau=-\frac{1}{3}\left( -e^{-t}+1\right)$
This should get you started. Good luck.