I think Dave's video does it as well as any of the others I have seen. I like the KCL approach - it brings a little formality to the process where handwaving doesn't. Both approaches get the same answer.

The Dan Bullard video is quite good in that he keeps working that awkward feedback circuit.

Where we don't have videos worth much is for the topic of analog computing. That is the original purpose for the op amp and those videos I watched this morning on solving differential equations just didn't do it (for me). Among other things, analog computers rarely use differentiators because they tend to inject noise (high pass filter) whereas integrators tend to remove noise (low pass filter).

I remember talking about the equation for the Mass-Spring-Damper problem in grad school and we set up the equation easily (well, most did...). But doing anything with it was maddening because the personal computer hadn't been invented. We certainly didn't have Matlab.

With real op amp integrators (2) and an op amp inverter plus a 2 input op amp summer, we can see the system work on an oscilloscope. In my case the integrators have a time constant of 1 so they work in real time. The feedback capacitor is 1 ufd and the input resistor is 1 Mohm. Oh, and there are a couple of potentiometers involved to set the constants and a scheme for setting the initial displacement (there is a force pulling down on the spring before it is released).

I have had more fun with differential equations in the last year than I ever had in college. In undergrad, the HP 35 hadn't even been invented!

Just for giggles, I have attached a PDF of the Matlab version. Matlab does a terrific job with this type of problem!

Op amps are fun!