The way I was taught about a characteristic equation is essentially "try plugging in $e^{ \lambda t}$ and see what happens". But is that really all there is to it? That doesn't intuitively lead me to more advanced techniques like taking a matrix exponential and Jordon-blocking it for more complicated circumstances.
Is there a more formally generalized process to explain what is going on when you derive a characteristic equation that would allow you to derive non-polynomial characteristic equations like $ \sin( \lambda) - \log(t)$ just as an example? Or what about other "characteristic" equations where you try plugging in $ \lambda \log(t)$? Why is that also not something that can generate a characteristic equation? Can the process be generalized to account for other functions besides $e^{ \lambda t}$?
Of course you can try to generalize this idea. If you can find a function family $\phi(t,\vec C)$, and inserting this into the differential equation allows you to find expressions in the constants in $\vec C$ that do not depend on the time $t$, that when setting to zero reduce the equation to a triviality, then you can also call these conditions on $\vec C$ characteristic equations.
But in general this will be very hard to do, compared with the success of the family of exponential functions $\vec ve^{λt}$, $\vec C=(λ,\vec v)$, for linear DE with constant coefficients.