RLC (ResistorInductorCapacitor) circuits are a common way of extracting a frequency range. Here's part of a typical circuit:
The middle of the schematic is the part we will be analyzing. It emphasizes one band of frequencies over the others, as we will see, and the potentiometer (refer back if you've forgotten the potentiometer) on the right acts as a volume control for the that band of frequencies (V_{out} will be a fraction of V_{R} controlled by the setting of the potentiometer wiper). A graphic equalizer contains many such circuits, each tuned to emphasize a different band. The output of all the channels are recombined after another stage of amplification, shown as the triangle on the right. In this way it separates the frequency bands, allows you to adjust the volume of each one individually, and then recombines them and sends the result to your speakers.
A timevarying voltage comes from your CD player, which is the sound waveform.The yellow elements represent stages of amplification, the one on the left being connected at its input end (left side) to youur CD player. We will treat the output of the one on the left as a voltage source measured with respect to ground (which we will name V_{in} because it is input to the part of the circuit under study). The input of the amplifier on the right we will assume to be an "observer" only: no current flows into it, but it watches the voltage V_{out} from our circuit and amplifies it. This concept is familiar to you already, because it's what voltmeters do.Thus it does not affect the behavior of the part of the circuit we are studying.
In analyzing circuits you have to be willing to focus on a chunk of it, and say "I can understand this part on its own, and I'll neglect its interaction with other parts". You can't generally write down or solve the equations of motion for a whole circuit at once. Here we don't know how the yellow amplifier blocks work, but we'll take the left one to be a voltage source and the right one to be a zerocurrent observer (and amplifier) of voltage.
Typically, coming in from the left is an audio signal, which is a mishmosh of different frequencies with different amplitudes, like this, where the horizontal axis is time (maybe milliseconds) and the vertical axis is volateg, Vin..
Let's consider instead a pure tone; a "sinewave" test signal, of the form
V_{in} = sin(wt) 
(2)

Now we'll find the equations of motion of the circuit. We expect to find two coupled diffeqs, becasue there are two "energy storage" elements: the inductor and the capacitor.
The constitutive relations of the elements are:
V_{R} = R i_{R} 
(3)
(4)
(5)

V_{in} = V_{L} + V_{C} + V_{R} 
(6)
(7)

State Variables
We need one state variable for each capacitor and each mass, thus in this system we will have two state variables. We could choose q_{C} and i_{L}, however q_{C} is actually inconvenient and instead we choose the voltage of the capacitor, V_{C}, which is linearly related to q_{C} via (5). Thus, i_{L} and V_{C} are our state variables, and we will try to eliminate all the other dynamic variables except V_{in}, which is an input. We quickly obtain these state equations:
i'_{L} = (1/L) (V_{in}  V_{C}  R i_{L}) 
(8)
(9)

We could now solve these diffeqs numerically. That will be homework. Here let's solve them analytically.
Noting that V_{in} is given by a sine in equation (2), I'm going to make these guesses for i_{L}(t) and V_{C}(t):
i_{L} = A sin(wt) + B cos(wt) 
(10)
(11)


(12)
(13)

Remember, the input is V_{in} = sin(wt), and the output (if the potentiometer is set to max) is V_{R}=Ri_{L}. We can put different sinewave test signals in by varying w.
So, how is this working? You will see in more detail in the Homework, but in the next section we will think about plotting V_{R} as a function of time, considering different frequency inputs. What we will find is that the amplitude of the output is a function of frequency, with the peak value occurring at a particular resonant frequency. That resonant frequency is determined by the values of the inductor and capacitor in the circuit. Thus, each channel of our graphic equalizer has a different capacitor and inductor, such that the frequency it controls is a particular range.