Ygggdrasil

You forgot the constant in the solution to the homogeneous equation:
X_H = Ae^t

Therefore the general solution to the non-homogeneous 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) = 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.