Impulse/Delta Function--homework help I need to solve the initial value problem:
I took the Laplace transform of both sides and this is what I have thus far:

I now need to take the inverse Laplace transform to find x(t). I can't simplify the denominator by completing the square, so I am stuck here. Is this an example of Duhamel's principle? If so, how do I solve that? Any help would be greatly appreciated! 

What you have looks correct to me. Now decompose the fraction this way:
$$G(s)=\frac 1 {(s^2+s-2)}=\frac 1 {(s+2))(s-1)}$$
$$G(s)=\frac 1 3 \left (\frac 1 {(s-1)}-\frac 1 {(s+2)} \right )$$
Then take Inverse Laplace Transform.
$$g(t)=\frac 13 (e^{t}-e^{-2t})$$
Do the same for the other fraction. 
