It looks like the system is
Compute the eigenvalues of the coefficient matrix.
For , the corresponding eigenvector is such that
Notice that the first row is 1 + 2i times the second row, so
Let ; then , so that
The eigenvector corresponding to is the complex conjugate of .
So, the characteristic solution to the homogeneous system is
The characteristic solution contains and , both of which are linearly independent to and . So for the nonhomogeneous part, we consider the ansatz particular solution
Differentiating this and substituting into the ODE system gives
Then the general solution to the system is