Impulse response function & Laplace transforms

[magnifik]

i am given the Laplace transform of an impulse response function, as well as its input. i am supposed to find its output.

H(s) = 1/s² + s + 1
x(t) = sin2(t-1)U(t-1)

what i have done so far is the following:
i know that Y(s) = H(s)X(s) and from this i can easily find y(t)
so i found X(s) since H(s) is already given...this may be wrong
X(s) = 2e^-s/s² + 4

then i found Y(s)
Y(s) = 2e^-s/(s² + s + 1)(s² + 4)
then i know that y(t) should be 2y(t-1) because of the e^-s time shifting

but i am stuck here... is there a better way to solve this problem using Convolution integral or any other way? i know how to do both the partial fractions method and convolution integral but i keep getting stuck. i think i have set up the problem wrong, perhaps in the X(s) solution

 

[rootX]

Yes it will be messy all ways!

 

[vela]

If you want to continue with what you started, ignore the exponential factor for now and find the inverse Laplace transform of

$$\frac{2}{(s^2+s+1)(s^2+4)}$$

Start by using partial fractions to get separate terms:

$$\frac{2}{(s^2+s+1)(s^2+4)} = \frac{As+B}{s^2+s+1}+\frac{Cs+D}{s^2+4}$$


The more painful way of finding y(t) would be to find h(t) from H(s) and then convolve x(t) and h(t).