The topic of this problem is The Complete Response of RLC Circuits.

The problem is to find the 2nd order differential equation expression for

the voltage, Vc(t) in the circuit shown below.

So, we have a circuit that has a series combination of R, Ls and Cs.

It also has a voltage source, VS sub t.

The voltage Vc(t) is the voltage that's across

the compositor in the top of our circuit.

So we're going to use Kirchhoff's voltage law, and

sum up the voltages around our single loop circuit.

To find an expression which will ultimately give us our route

to finding our second order differential equation for Vc(t).

So if we use Kirchhoff's voltage law, it's a single loop circuit,

so only one loop to choose from, and we're going to travel

clockwise around our circuit starting in the lower left corner.

The first thing we encounter is a negative polarity of the voltage source Vs(t).

As we continue around this, we run into the inductor and

the voltage across the inductor.

We know that the voltage across

the inductor is L di, dt.

And it's an I of t it's a function of time,

and this is our single loop current I of t.

We also have our voltage Vc(t), it's part of this expression

as well because we capture the capacitance next and then we have the resistor.

And the voltage throughout across the resistor is R times i(t),

and so the sum of those is equal to 0.

So if we use our well known expression for the capacitor that is,

that the current through the capacitor is equal to the capacitance times,