For every complex number $r$ and integer $n$, let $e_{r,n}:x\mapsto x^n\mathrm e^{rx}$. If $r$ is a root of multiplicity $k$ of the characteristic equation, then the function $e_{r,n}$ is a solution for every $0\leqslant n\leqslant k-1$. The family of functions $(e_{r,n})_{r,n}$ is linearly independent hence this yields a vector space of solutions whose dimension is the degree of the equation. That is, the exact and complete set of solutions.
Let $c_{t,n}:x\mapsto x^n\cos(tx)$ and $s_{t,n}:x\mapsto x^n\sin(tx)$. The formula in your post describes the general element of the vector space generated by $e_{0,0}$, $e_{0,1}$, $e_{\mathrm i,0}$, $e_{\mathrm i,1}$, $e_{-\mathrm i,0}$ and $e_{-\mathrm i,1}$ since, for every real number $t\ne0$, $e_{\mathrm it,n}$ and $e_{-\mathrm it,n}$ generate the same vector space as $c_{t,n}$ and $s_{t,n}$.
Finally, there exists some coefficients $\lambda_n$, $\mu_n$ and $\nu_n$, for $n=0$ and $n=1$, such that $ y=\lambda_0e_{0,0}+\lambda_1e_{0,1}+\mu_0c_{1,0}+\mu_1c_{1,1}+\nu_0s_{1,0}+\nu_1s_{1,1}. $