Take the Laplace transform of the equation:

s^2 Y(s) -sy(0) - y'(0) -2[s Y(s) - y(0)] + 2Y(s) = 8 (s+1)/[(s+1)^2+1]

(s^2 -2s +2)Y(s) = 8 (s+1)/[(s+1)^2+1] + 16s - 16

So (if I have done this right)

Y(s) = 8(2s^3 + 2s^2 + s - 3)/[(s^2 + 2·s + 2)(s^2 - 2·s + 2)].

To finish this expand the left hand side in partial fractions and look up the

corresponding inverse LTs to get the solution.

RonL