In the next three videos, I want to show you some nice applications of these second-order differential equations. The first one is from electrical engineering, is the RLC circuit; resistor, capacitor, inductor, connected to an AC current with an EMF, E of t. To write down the differential equation here, we need the constitutive relations for these circuit elements. So, I will just write them down. So, the voltage drop across a resistor, is equal to the current in the circuit times the resistors. The voltage drop across a capacitor, is equal to the charge on the capacitor divided by the capacitance C, and the voltage drop across the inductor, is equal to the derivative of the current with respect to time, times L. So, these define essentially the circuit elements, and then there's a charge on the capacitor and the current in the circuit. Those are related by the current is equal to the time derivative of the charge. How do we get a differential equation? The differential equation comes from Kirchhoff's Law, which says that, "the voltage supplied by the battery, or here it's the AC voltage epsilon of t, or Eft, is equal to the voltage drop across each of the circuit elements when they are in series." So, here let me write that as V_L plus V_R plus V_C. Because these circuit elements have derivatives in them, this is a differential equation. The AC current can be modeled as E of t equals some amplitude e naught times cosine Omega t, okay? So, that's a sinusoidal AC current. So, putting this together, what is the differential equation? So, V_L is, L times di/dt, but i is dq/dt. So, we can write a differential equation in terms of Q. So, VL becomes L, d squared q dt squared. VR is i times R, i is dq/dt. So, plus VR becomes plus R dq/dt, and then VC is just q over C, plus q over C. That's equal to the AC current which is E naught times cosine omega t, okay? That's our differential equation. You should recognize this L, R, C, E naught are parameters that are constant. Omega is a constant. So, this is a second-order linear differential equation in homogeneous with constant coefficients. So, that's exactly what we've been studying. This equation has a lot of parameters in it, L, R, C, E naught omega. It pays to reduce the number of parameters. If you divide through by L, you see this term here has a q over LC. If you remember our oscillator equation x double.plus omega naught squared x equals some force, then the term omega naught squared is the one. When this becomes d square qdt squared, the term omega naught squared is the term multiplying q, and here it's equal to one over LC. So, we can define an omega naught equal to one over the square root of LC, and that's the natural frequency of this RLC circuit when you don't have a resistor, when there is no damping. This has units of one over time. So, we can non-dimensionalize this equation to give us an equation with fewer parameters. We can use this omega naught, which has units of one over time, to non-dimensionalize time. So, we can define the non-dimensional time tau equals to omega naught times t. Then, tau is unit-less, t has units of whatever units of time, but now we've non-dimensionalized it. So, when tau goes from zero to one, t is going from zero to one over omega naught. So, in some sense we've chosen one over omega naught as our unit of time. Okay. Then, we can also non-dimensionalize q. You can play with this equation and try to determine how do you define the charge q, so that in the non-dimensional equation this term will clear. I won't do that here, but I can write it down. We define the dimensionless time q by capital Q, and that will be omega naught squared L, divided by E naught times the charge q. This has units of one over charge times charge, and then we'll get a dimensionless charge. You take these definitions, and substitute it into the differential equation, and you end up with a dimensionless equation, which looks like d squared Q over d Tau squared, plus alpha dq/dTau, plus Q equals cosine beta Tau. Where you have two non-dimensional parameters in this equation, the alpha and the beta, alpha is given by R over L omega naught, and beta is given by omega divided by omega naught, and these are dimensionless. So, we went from a full dimensional equation for the RLC circuit. By redefining dimensionless variables, we end up with an equation that no longer has dimensions. Each term is unit-less, and we end up with two parameters, alpha and beta. Okay. Let me review here. We have the RLC circuit which is a simple circuit from electrical engineering with an AC current. If we want to write down the differential equation for this circuit, we need the constitutive relations for the circuit elements. This defines what it means to be a resistor, a capacitor, and an inductor. There is a relationship between current and charge through the derivative. Then, we write down Kirchhoff's Law, which is E of t is equal to the voltage drop across the resistor plus the voltage drop across the capacitor, plus the voltage drop across the inductor. That gives us our second order differential equation. We want to define something, a parameter that has units of one over time. We can do that by dividing through by L, and picking up the one over LC times the q term. One over LC is our omega naught squared. So, we can define the natural frequency of the circuit without a resistor, is one over root LC, make a non-dimensional time, a non-dimensional charge, and we end up with this second-order differential equation with two parameters. It's interesting, but we're going to see that we're going to discuss two more applications, completely different physical problems, but we are going to end up with exactly the same equation in the end, okay? So, that will be interesting. The only thing different will be the definitions of these dimensionless parameters. So, let's continue with the next two videos, and see two more applications. I'm Jeff Jasanoff, thanks for watching, and I'll see you in the next video.