You forgot the constant in the solution to the homogeneous equation:
XH = Aet
Therefore the general solution to the non-homogeneous ODE is:
x(t) = Aet + (1/2)(sin(t)-cos(t))
To find the value for A, you should use the boundary condition x(0) = x0. (You should find that A = x0 + 1/2).
Alternatively, you could carry the constant through your original solution and solve for C using the boundary condition.