To compute generalized eigenvectors of defective linear ODE systems, it is up to the mathematician's trial and error to find and plug in the specific eigenvector(s) which are in the column space of the system. This is how solutions multiplied by successive powers of $t$ are found for systems. The analogous method for finding missing solutions to single ODEs is Reduction of Order. Given how finding generalized eigenvectors works, I expected that for higher order systems, the correct fundamental solution would have to be plugged into Reduction of Order for the algorithm to work. However, this turned out to be false.
The problem I tried was $y''' - 4y'' - 3y' + 18y = 0$. The first two fundamental solutions $e^{-2t}$ and $e^{3t}$ can be found via the characteristic polynomial. Although it's clear by this point that $3$ is the eigenvalue with defective algebraic multiplicity, I performed Reduction of Order using $y = ve^{-2t}$, and found the correct solution, $y = Ae^{-2t} + Be^{3t} + Cte^{3t}$. I also got the correct solution through Reduction of Order using $y = ve^{3t}$, and even $y = vte^{3t}$.
How is Reduction of Order able to work, even when fed the "wrong" fundamental solution, while the method for finding generalized eigenvectors of systems is not?