You forgot the constant in the solution to the homogeneous equation:
X_{H} = Ae^{t}
Therefore the general solution to the nonhomogeneous ODE is:
x(t) = Ae^{t} + (1/2)(sin(t)cos(t))
To find the value for A, you should use the boundary condition x(0) = x_{0}. (You should find that A = x_{0} + 1/2).
Alternatively, you could carry the constant through your original solution and solve for C using the boundary condition.
