Calling the variable voltage source voltage f(t) and the system output voltage, y(t),
The problem asks that I derive a differential equation f(t) =... in terms of these functions and LL, RL, CL, CS.
So far, I've used KCL, where I = I(CL)+I(CS)...(1), which can be rewritten as dy(t)/dt * CL + dy(t)/dt * CS...(2).
After this I am unsure. I tried to find I initially with corresponding current equations for the resistor and inductor, but I got stuck.
Then I used KVL around the loop on the left, which gave f(t) = LL(dI/dt) + I(RL) + y(t).
Substituting equation 2 into I, I end up with:
f(t) = [LL(CL+CS) D^2 + RL(CL+CS)D + 1]y(t), where D is the operator for d/dt.
Is there something I'm missing? I am just pensive about my equation because the next question asks that I find the characteristic roots -- with LL=0.02, RL=0.50, CL=5.0, and CS=0.10, which end up being not-nice numbers for finding roots.