Analog computing

Integrator and Differentiator shown :





Regards, Dana.

Not sure if it solves a differential equation by being a differential detector. The op amp contains a long-tail pair which acts as the differential detector. An unchanging current is shared by two networks resembling class A amplifiers. If the left-hand column draws less Amperes, this allows the right-hand column to increase the amount of Amperes it draws. And vice-versa and conversely. The resulting network is not too different from a Wheatstone bridge. And the Wheatstone bridge concept is frequently extended to the input network.

To determine slope (rising or falling) of a signal, I've seen a capacitor integrator at one input of the op amp causes the output to go high or low. The signal is split so it goes unchanged to one input, and a slightly delayed copy goes to the other input. If the delayed copy is greater than the initial signal, then the output goes one direction. If the delayed copy is less than the initial signal, the output goes the other way.

I don't think the hand drawn circuit performs the equation.
Equation has Vi as "output", circuit has Vi as input. Maybe
if you put the whole thing inside a feedback loop, that
would have an output at Vi which truly did what the text
indicates.

The links in post #2 just ref material.

With some experimentation I found the proper amount of gain for my setup. Incoming signal reaches peak amplitude briefly. At the same moment output is at zero thus indicating slope of incoming signal and performing differential function.

A lissajous plot shows the relationship. Incoming signal is a sine wave (pretty much) therefore the differential looks almost like the cosine.

To be strictly correct a second op amp is needed to invert polarity of the result.

I was trying to understand the method to solve the second equation(#1), dc/dt=beta/L* n - lamda*C. The problem with this equation is that outputs do go unbounded due to presence of integrator, eventually saturating with the input being very small. Am I correct?

Yes, I had to fiddle with component values to keep the output within bounds.
The output tended to pin to the supply rails. (My input signal is Falstad's simulator ANTenna model whose amplitude varies all over the place. I took the screenshot while amplitude was large and getting larger.)

I guess it seems strange to put an integrating capacitor to solve a differential equation. There may be another topology which is more straightforward.