This question is inspired from [Andy Aka's question](https://electronics.stackexchange.com/q/373081/38098) from almost 5 years ago. It is also a nod of my respect to [Andy](https://electronics.stackexchange.com/users/20218/andy-aka). But what I wrote there is kind of muddled. And it only addresses the under-damped case. I'd like to end my time here with something better than what I provided then to Andy.
I've benefited a lot from some exceptional folks here, appreciating them still more over time. Some I have already missed have left for their own reasons this last decade. I will also sorely miss still more when I also depart in a few days (end of the month.) This is my last question and perhaps, if forced to it, then on December 31th also my last answer.
Assume the following schematic:
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The specified initial conditions are \$V_{_0}\$ and \$I_{_0}\$. The values of \$R_1\$, \$C_1\$ and \$L_1\$ are all non-zero and positive and constant over time. They are to be taken as ideal components, too (no parasitics to be added.)
There are three cases to deal with. Critically-damped, under-damped, and over-damped. There's no specification as to which of these apply, so there are potentially three different specific solutions.
I don't want general solutions with unspecified constants. Instead, I want to see the ***specific*** solutions for the voltage across the capacitor, using the initial conditions and part values.
The initial conditions apply at \$t=0\$ and the specific solutions for all three cases will be for \$t\ge 0\$.
Finally, the development must start with provided time-domain KCL (see below.) The development should be strictly performed using only time-domain development and must be relatively easy to follow for those familiar with no more than the first six chapters of the 9th edition of Nagle, Saff, and Snider's ***Fundamentals of Differential Equations***. (This excludes Laplace and/or inverse Laplace as that doesn't begin until chapter 7 in that textbook.) Methods such as *undetermined coefficients* or other more rigorous approaches are fine.
The problem is rather simple in the sense that there are no non-homogeneities (no driving functions.) I'll start the process with the KCL:
$$\begin{align*}
\frac{v_{_\text{C}}}{R_1}+C_1\frac{\text{d}}{\text{d}t}v_{_\text{C}}+\frac1{L_1}\int v_{_\text{C}}\:\text{d}t&=0\:\text{A}
\\\\
v_{_\text{C}\left(t=0\right)}&=V_{_0}
\\\\
i_{_\text{L}\left(t=0\right)}&=I_{_0}
\end{align*}$$
Clearly, this is an homogeneous equation. So it should be fairly straight-forward.
Positive currents are ***downward***.
I will not accept an answer where the results are just copied out without any development shown. There must be a visible step-wise process. I will write an answer if nothing acceptable is provided before the end of the year, as that's when I'm leaving the site and won't be returning. Hopefully, I will accept an earlier answer, though.
The ***specific*** solutions must cover all three possible cases mentioned above: critical-, under-, and over- damped. A simulation for each possible case must be included with the answer and at least a few points from each *specific* solution case equation shown to match up with the simulations of each case. In other words, simulation must confirm the specific solution equations.
For example, here are three curves:
[![enter image description here][1]][1]
The *red* curve is for an over-damped case, the *green* curve is for a critically-damped case, and the *dark blue* curve is for an under-damped case.
I think this passive formulation, combining one each of the two basic energy storage devices and avoiding complex driving functions, is an essential fundamental skill.
(I will add a bounty to the question on the 27th of December.)
**Note**: I forgot to add another constraint. The final equations (there will be three, I believe) must symbolically use \$R_1\$, \$C_1\$ and \$L_1\$ and they cannot use complex or imaginary numbers. They may have required the use of such in their development. But the final equations must be entirely real-in, real-out equations. More specifically, it's not allowed to use complex-valued roots as elements to a complex-compatible exponential function. Exponential functions are fine, so long as they are a real-in, real-out subset.
[1]: https://i.sstatic.net/HNUVs.png