In the undriven $RLC$ circuit, why would one conjecture the solution with $Ae^{Zt}$ when we perfectly know that critical case has $te^{-\alpha t}$?

Question

There's something I don't quite get in this video (MIT $8.02$ course titled "Electricity and Magnetism", video number $208$, taught by Pr. Peter Dourmashkin).

Professor conjectures the solution to the series, undriven $RLC$ circuit ODE by the function $Ae^{Zt}$ where $Z\in \mathbb C$.

But, as some of you may know, the solution to the ODE is, in the critically damped situation, of the form $\lambda_1 e^{-\alpha t}+ \lambda_2 te^{-\alpha t}$ (where $\lambda_i \in \mathbb R$ and $\alpha >0$ is the double root of the characteristic equation).

To what extent does his conjectured solution match the second term of the critically damped solution (i.e. $t\mapsto te^{- \alpha t}$)?

Answer 1

Conjecturing solution of the form $e^{\alpha t}$ allows us to find characteristic polynomial easily and then it is standard fact that in cases like this one, if $\alpha$ is repeated root, then general solution is combination of $e^{\alpha t}$ and $t e^{\alpha t}$.