What does it mean when my circuit has an indeterminate/no solution?
I did arithmetic until I ended up with a single equation with two unknowns 40 + v1 = v2.

This is precisely the goal of the problem author. They want to show you a problem that can't reasonably be solved when every source is ideal. There's a surprisingly rich set of ramifications from this, which affects real-world simulations that don't look like this contrived circuit.

I get the feeling that v1 and v2 may behave with resonant behavior or switch amplitudes interconnectedly as the resonant RLC circuits behave.

I won't concern myself with real-world circuits, because you have given a schematic model and we ought to solve it using tools appropriate to schematic models.
Resonant behavior (or amplitude fluctuation) is a product of a real-world circuit. Here, the sources are ideal mathematical models. The problem is purely mathematical, and we would only be able to conclude that we cannot find a full meaningful solution.
For this specific circuit, there is an infinite family of possible solutions, described by the equation 40 + v1 = v2, because there is another degenerate constraint (the currents flowing through the left and center branches happen to exactly sum to 20 mA).
In terms of the language used by circuit simulator error messages, nodes V1 and V2 are said to &quot;float&quot; relative to ground. An analogy would be trying to solve this system of linear equations:
$$ \begin{align}x + y &amp;= 1\\
0 &amp;= 0\end{align}$$
Now suppose that we tweak one current source by a tiny amount. This could be a numerical or rounding error in a simulation model, or just by design:



simulate this circuit &ndash; Schematic created using CircuitLab
This circuit is still degenerate, but now it has no possible solution. Can you see why?
(I strongly urge you to stop reading here and try to work it out for yourself on paper if it isn't immediately obvious).
In our equation analogy, we are now solving a structure that looks mathematically like:
$$ \begin{align}x + y &amp;= 1\\
0 &amp;= 0.01\end{align}$$
No matter what you do with \$x\$ and \$y\$, there's just no way to make 0 equal to 0.01. Same here. No matter how much you play with voltages, you're not going to be able to fix the fact that 20.001 mA of current enter node V2, while only 20 mA of current leave it.
This is one reason why circuit simulators use slightly non-ideal sources and models. One common option is a Gmin, where a large (i.e. gigaohms or higher) resistor is inserted (automatically by the simulator) in parallel with certain problematic elements:

Can you now solve this circuit by hand? You can. The values may be absurd (i.e. kilovolts present at the nodes), but at the very least the model is solvable (in simulator parlance, it is said to converge now)
(Again, I recommend that you stop here, and try to work out the circuit on paper. You don't need to necessarily solve it given the annoying numbers, but convince yourself that it does have a unique, well-defined solution. As a further exercise, take a circuit which can be solved and has reasonable values for all its components, add some GMins, and show that the answer is approximately equal to that without the GMin terms)
The circuit shown here is a contrived example, since it's always ill-conditioned, but there are real-world circuits that, for one reason or another, end up ill-conditioned like this for brief moments as they settle (this is an artifact of the simulation, and the circuits being simulated never blow up like this in real life). Imagine for instance, a common CMOS inverter (taken from The Designer's Guide to Spice and Spectre by Kundert1):

You could have a model for the circuit which makes the upper and lower transistors have infinite output impedance (i.e. acting like ideal current sources) in certain conditions. For this answer, we don't really need to concern ourselves with what a transistor is, what its model is composed of, or how it varies in all circumstances. All we care about is that the model breaks down with a floating node (like above), in one particular set of circumstances.
That's not a model that I would ever prefer to use for a variety of reasons, but let's assume that we have this crappy model and are stuck with it. The chart from the same figure shows the issue:

When the input voltage is in the middle, there's a whole family of solutions and the output voltage could be literally anything. Like before, we can throw GMin to make the infinite slope not quite infinite, thus giving us a hope of convergence.
A side note: I appreciate the desire to post a Youtube video, but this is a pretty rich topic with a variety of pitfalls and misconceptions, and it's very easy to accidentally say the wrong thing or misinterpret the math, even with the best of intentions.
I strongly suggest that you take the time to really understand the analysis going on, as well as things such as analyzing impedances &quot;seen&quot; by nodes and elements, and so on, before trying to teach this topic to others. You should ideally be deeply familiar with solving real-world circuits, including non-linear ones, to better understand not only the material, but what to emphasize and the pitfalls that may befall you or your viewers.
1 I highly recommend this book if you're interested in simulation. It discusses various simulation pitfalls and how they are handled by the Spice and Spectre simulation engines. Kundert was affiliated with Cadence Design Systems so he does hint at the benefits of Spectre over Spice quite often, but I am taken to agree with them.