Consider a homogeneous linear system of differential equations $x' = Ax$ with $A$ a real $7 \times 7$ matrix where we only know the eigenvalues and their multiplicities: $\lambda_1 = 2$ with multiplicity $m_1 = 3$, $\lambda_2 = 0$ with $m_2 = 2$ and complex conjugated solutions $\lambda_3 = 1+i$ and $\overline \lambda_3 = -1 - i$. How do we determine a basis of the solution space of this system?
Edit: From a known proposition one knows that our basis has the form $e^{2t} p_0(t), e^{2t} p_1(t), e^{2t} p_2(t)$ (solutions associated to $\lambda_1$), $q_0(t), q_1(t)$ (solutions associated to $\lambda_2$) and $e^{-t}(\cos (t) r_0(t) - \sin (t) s_0(t)), e^{-t}(\cos (t) s_0(t) + \sin (t) r_0(t))$ (solutions assoc. to $\lambda_3$). Where the subindex of the $p_j$ etc gives the maximal degree of the respective polynomial function. What I don't know is how to determine the coefficients of these polynomials.