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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.

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 31st also my last answer.

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.

The specific solutions must cover all three possible cases mentioned above: critical-, under-, and over- damped. A simulation (the only time numeric values are allowed, as that's needed in order to demonstrate that the solutions deduce into specific situations) 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.

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 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.

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.