This question is inspired from Andy Aka's question from almost 5 years ago. It is also a nod of my respect to Andy. 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:

^{simulate this circuit – Schematic created using CircuitLab}

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.

To expand on the above and set the tone for the development:

$$\begin{align*} \frac{v_{_\text{C}}}{R_1}+C_1\frac{\text{d}}{\text{d}t}v_{_\text{C}}+\frac1{L_1}\int v_{_\text{C}}\:\text{d}t&=0\:\text{A} \\\\ \frac{\text{d}}{\text{d}t}\left[\frac{v_{_\text{C}}}{R_1}+C_1\frac{\text{d}}{\text{d}t}v_{_\text{C}}+\frac1{L_1}\int v_{_\text{C}}\:\text{d}t\right.&=\bigg.0\:\text{A}\bigg] \\\\ \frac1{R_1}\frac{\text{d}}{\text{d}t}v_{_\text{C}}+C_1\frac{\text{d}^2}{\text{d}t^2}v_{_\text{C}}+\frac{v_{_\text{C}}}{L_1}&=0\:\text{A} \\\\ \frac{\text{d}^2}{\text{d}t^2}v_{_\text{C}}+\frac1{R_1\,C_1}\frac{\text{d}}{\text{d}t}v_{_\text{C}}+\frac{1}{L_1\,C_1}v_{_\text{C}}&=0\:\text{A} \\\\ \left[\frac{\text{d}^2}{\text{d}t^2}+\frac1{R_1\,C_1}\frac{\text{d}}{\text{d}t}+\frac{1}{L_1\,C_1}\right]v_{_\text{C}}&=0\:\text{A} \\\\ \left[D^2+\frac1{R_1\,C_1}D+\frac{1}{L_1\,C_1}\right]v_{_\text{C}}&=0\:\text{A} \end{align*}$$

To zero out the left side above there are two choices. One is to set \$v_{_\text{C}}=0\$. But given that the problem allows the specification of \$V_0\ne 0\$ and \$I_0\ne 0\$, that option is readily disposed of.

So there's only one other choice: \$\left[D^2+\frac1{R_1\,C_1}D+\frac{1}{L_1\,C_1}\right]=0\$. That's just a matter of solving for its quadratic roots.

Or, it would be if I were permitting complex/imaginary roots to be used in the final solutions. But I'm disallowing those in the final answers. So, while the development may use mathematical properties, including complex domain ones such as Euler's, the final equations must be free of complex/imaginary values and must be entirely symbolic, in form.

I will not accept an answer where the results are just copied out without any development shown. For example, if the critically-damped solution is simply "copied out of a book." At least show some development to get there. 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:

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: The final equations (there will be three, I believe) cannot use complex or imaginary numbers. They should symbolically use \$R_1\$, \$C_1\$ and \$L_1\$. You may use Euler's in their development, of course. But in the end the final equations must be entirely real-valued in, real-valued out solutions. Exponential functions are fine and encouraged, so long as they are a real-valued in and real-valued out. And don't forget that I'm looking to see their development from start to finish, using the symbols for the three parts and no values specified. No 'skipping over incremental development steps' by using inputs to and resulting outputs from something like Wolfram Alpha, for example. In the end, of course, values will have to be supplied to verify. But no rational-real values may be used during the development of the symbolic solutions.

An answer should make a point of illustrating how we know what we know. The better it does that the more likely I select it.