I am trying to get a better understanding of how oscillator circuits (clapp, hartley, colpitts etc) work, so I have been trying to solve the differential equation for a very simple one--the one most the way down the page here: http://www.electronics-tutorials.ws/oscillator/oscillators.html.

I can do the diffy-q for the LC and RLC circuits, but as soon as I add in a bigger circuit my equation breaks down, which I assume is due to my model for the npn transistor:

-Base-Emitter voltage drop is 0.6 v

-Collector Current is Beta times Base Current

-Work around Collector-Emitter voltage since I don't have a relationship for that

I feel like those assumptions work for steady state dc circuits, but not for this time-domain analysis. Is there a more complicated model that allows for the differential equations?

I ideally I would like to solve the circuit with with a small five-ohm speaker attached to V-out so I can guess the frequency of the sound and compare to a guitar tuner... But I'm a little stuck at the transistor.

Thanks for looking!